{"text":"\\section{Introduction}\\label{sect1}\n\n \\section{Introduction}\\label{sect1}\n\nWe consider random combinatorial objects which can be described in\nterms of their component structure.  For an object of weight $n$, denote\nthe component structure by \n$$\n\\b{C} \\equiv \\b{C} (n) \\equiv (C_1(n),C_2(n),\\ldots,C_n(n)),\n$$\nwhere $C_i \\equiv C_i(n)$ is the number of components of size $i$. Since\n$i C_i$ is the total weight in components of size $i$, we have \n$$\nC_1 + 2C_2 + \\cdots + n C_n =n.\n$$\nFor each fixed $n$, by choosing an object of weight $n$ at random, with\nall possibilities equally likely, we view $\\b{C} (n)$\nas a $\\B{Z}_+^n$-valued stochastic process, whose coordinates $C_i(n),\ni=1,\\ldots,n$, are {\\it dependent}, nonnegative integer--valued \nrandom variables.  \nThis paper considers combinatorial objects for\nwhich the joint distribution of $\\b{C}(n)$ can be expressed as the joint\ndistribution of {\\it independent} random variables $Z_1, Z_2, \\ldots, Z_n $\nconditioned on the value of a particular weighted sum.\n\n    There are at least three broad classes of combinatorial structures\nwhich have this description in terms of conditioning an independent\nprocess.  The first class is assemblies of labeled structures on\n$[n] \\equiv \\{1,2,\\ldots,n \\}$; see Joyal (1981). This class includes  \npermutations, decomposed into cycles; mappings, decomposed into\nconnected components; graphs, decomposed into connected components, \nand partitions of a finite set.  The second class\nis multisets, i.e. unordered samples taken with replacement. This class\nincludes \npartitions of an integer; random mapping patterns; and monic polynomials\nover a finite field, decomposed into monic irreducible factors.  The\nthird class is selections, i.e. unordered samples taken without replacement, \nincluding partitions of an integer into parts of distinct sizes, and\nsquare-free polynomials.\n\n    The detailed description of any of the above examples is given in\nterms of a sequence of nonnegative integers $m_1, m_2, \\ldots $ \\  .\n For assemblies, let $m_i$ be the number of labelled structures \non a set of size $i$, for $i=1,2,\\ldots $ ;\n permutations have $m_i =(i-1)!$,  mappings have\n$ m_i = (i-1)!(1 + i + i^2 \/2 +... + i^{i-1}\/(i-1)!)$, and partitions\nof a set have  $m_i=1$.\nFor multisets and selections, let $m_i$ be the number of objects of\nweight $i$; partitions of an integer have $m_i=1$,\nand the factorizations of monic polynomials over a finite field have $m_i$\nequal to the number of monic, irreducible polynomials of degree $i$.\n\nFor $\\b{a} \\equiv (a_1,a_2, \\ldots, a_n) \\in \\B{Z}^n_+$, consider the\nnumber  $N(n,\\b{a})$ of\nobjects of total weight $n$, having $a_i$ components of size $i$, for $i\n= 1$ to $n$.  For assemblies, the generalization  of Cauchy's formula for\npermutations is the enumeration\n\\begin{eqnarray}\nN(n, \\b{a}) & \\equiv &  | \\{ \\mbox{assemblies on $[n]$}: \\b{C} = \\b{a} \\} |\n\\nonumber \\\\\n& = &\n \\mbox{\\bf 1}(a_1+2 a_2 + \\cdots + n a_n =n) \n\\   n! \\ \\prod_1^n \\frac{m_i^{a_i}}{(i!)^{a_i} \\  a_i !}.\n\\label{Nassembly}\n\\end{eqnarray}\nFor multisets, \n\\begin{eqnarray}\nN(n, \\b{a}) & \\equiv &  | \\{ \\mbox{multisets of weight $n$}: \\b{C} = \\b{a}\n\\} |\n \\nonumber \\\\\n& = &\n \\mbox{\\bf 1}(a_1+2 a_2 + \\cdots + n a_n =n) \n\\    \\ \\prod_1^n {m_i +a_i -1 \\choose a_i }.\n\\label{Nmultiset}\n\\end{eqnarray}\nFor selections, \n\\begin{eqnarray}\nN(n, \\b{a}) & \\equiv &  | \\{ \\mbox{selections of weight $n$}: \\b{C} = \\b{a}\n\\} |\n  \\nonumber \\\\\n& = &\n \\mbox{\\bf 1}(a_1+2 a_2 + \\cdots + n a_n =n) \n\\    \\ \\prod_1^n {m_i  \\choose a_i }.\n\\label{Nselection}\n\\end{eqnarray}\n\n\nLet $p(n)$ denote the total number of structures of weight $n$, to wit \n\\begin{equation}\\label{def p(n)}\np(n) = \\sum_{\\b{a} \\in \\B{Z}^n_+} N(n,\\b{a}).\n\\end{equation}\nFor permutations, $p(n)=n!$; for mappings, $p(n)=n^n$; for graphs, $p(n)\n= 2^{n \\choose 2}$; for partitions of\na set $p(n) = B_n$, the Bell number; for partitions of an integer,\n$p(n)$  is the standard notation; and for monic polynomials over a field\nwith $q$ elements, $p(n)=q^n$. \n\n\n\n\nA {\\it random structure} is understood as follows. Fix a constant $n$, and\nchoose one of the $p(n)$ structures at random, with each possibility\nequally  likely.  This makes $\\b{C}(n)$ a\nstochastic process with values in $\\b{Z}_+^n$, whose distribution is\ndetermined by \n\\begin{equation}\\label{combdist.old}\n\\B{P}(\\b{C}(n)=\\b{a}) \\equiv \\frac{N(n,\\b{a})}{p(n)},\\ \\b{a} \\in\n\\B{Z}_+^n.\n\\end{equation}\n\nIn Section \\ref{sect2} below, we show that there are independent random\nvariables $Z_1, Z_2, \\ldots$ such that\nthe combinatorial\ndistribution (\\ref{combdist.old}) is equal to the joint distribution of\n$(Z_1,Z_2,\\ldots,Z_n)$ conditional on the event\n$\\{T_n =n \\}$, where \n$$\nT_n \\equiv Z_1+2Z_2+\\cdots+nZ_n.\n$$\nExplicitly, for all \n$\\b{a} \\in \\B{Z}_+^n$\n\\begin{equation}\\label{equaldist}\n\\B{P}(\\b{C}(n)=\\b{a}) = \\B{P}\\left( (Z_1,Z_2,\\ldots,Z_n)=\\b{a} \\left|\n T_n=n\n\\right. \\right).\n\\end{equation}\n\nThe distribution of the degrees of the factors of the characteristic \npolynomial of a uniformly chosen random\nmatrix  over a finite field can also be expressed as in\n(\\ref{equaldist}) (cf.\nHansen and Schmutz (1994)), but it is not an example of an assembly,\nmultiset, or selection.\n\n  It is fruitful to compare the combinatorial structure directly\n to the independent discrete process, without renormalizing. The quality of\n approximation can  be usefully quantified in terms of total variation\n distance between the restrictions of the dependent and independent\nprocesses to a subset of the possible coordinates. We carry this out in\nSection \\ref{sect3}. \nBounds and limit theorems for natural functionals which depend on the\ncoordinates, albeit weakly on those outside a subset, are then easily\nobtained as corollaries. For examples of this in the context of random\npolynomials over finite fields, and random\npermutations and random mappings, see Arratia, Barbour, and Tavar\\'e\n(1993), and Arratia and Tavar\\'e (1992b).\n\n\nThe comparison of combinatorial structures to independent processes,\nwith and without further conditioning, has\na long history.  Perhaps the best known example is the representation of\nthe multinomial distribution with parameters $n$ and $p_1,\\ldots,p_k$ as\nthe joint law of independent Poisson random variables with means\n$\\lambda p_1, \\ldots, \\lambda p_k$, conditional on their sum being equal\nto $n$.  \n\n\nHolst (1979a) provides an approach to urn models that unifies \nmultinomial, hypergeometric and P\\'olya sampling. The joint laws of the\ndependent counts of the different types sampled are represented,\nrespectively, \nas the joint  distribution of independent Poisson,\nnegative binomial, and binomial random variables, conditioned\non their sum. See also Holst (1979b, 1981). The quality\nof such approximations  is assessed using metrics, including the total\nvariation distance, by Stam (1978) and Diaconis and Freedman (1980).  \n\nThe books by Kolchin, Sevast'yanov, and\nChistyakov (1978) and Kolchin (1986) use the representation of\ncombinatorial structures, including random permutations and random\nmappings, in terms of independently\ndistributed random variables, conditioned on the value of their sum.\nHowever, the Kolchin technique requires that the independent variables be {\\em\nidentically} distributed.  The\nnumber of components $C_i$ of size $i$ is the number of random variables\nwhich take on the value $i$.\n\nShepp and Lloyd (1966) study random permutations using\n a conditional relation almost identical\nto (\\ref{equaldist}), with ${\\Bbb E} Z_i=x^i\/i$ and $x=x(n)$, \nexcept that they condition on $n$ being the value\nof an infinite sum $Z_1+2Z_2+\\cdots$, which of course entails that\n$Z_{n+1}=Z_{n+2}=\\cdots=0$, and requires $x<1$. Variants on the\n Shepp and Lloyd technique are discussed by Diaconis and Pitman (1986),\nand are effectively exploited to prove functional\ncentral limit theorems for two combinatorial assemblies by Hansen (1989,\n1990), and used as a convenient tool for moment calculations by Watterson\n(1974a) and Hansen\n(1993). A related technique, coupled with an observation of Levin (1981), is\nused by Fristedt (1992, 1993) to study random partitions of a set and\nrandom partitions of an integer.\n\n\\subsection{Notation}\n\nThere are several types of asymptotic relation used in this paper. For\nsequences $\\{a_n\\}$ and $\\{b_n\\}$, we write\n $a_n \\sim b_n$ for the asymptotic relation\n $a_n\/b_n \\to 1$ as $n \\to \\infty$. We write\n $a_n \\bothsides b_n$ if there are constants $0 < c_0 \\leq c_1 <\n\\infty$ such that $c_0 b_n\n\\leq a_n \\leq c_1 b_n$ for all sufficiently large $n$. \nWe write $a_n \\approx b_n$ to denote that $\\log a_n \\sim \\log b_n$. \nFinally, we say that $a_n \\doteq b_n$ if $a_n$ and\n$b_n$ are approximately equal in some heuristic sense\ndeliberately left vague.\n\nFor $r \\in {\\Bbb Z}_+ \\equiv \\{0,1,2,\\ldots \\}$,\nwe denote the rising factorial $y_{(r)}$ by $y_{(0)} = 1, \ny_{(r)} = y(y+1) \\cdots (y+r-1)$  and the falling factorial $y_{[r]}$ by \n$y_{[0]} = 1, y_{[r]} = y(y-1) \\cdots (y-r+1)$.\nWe also write ${\\Bbb N} \\equiv \\{ 1,2,\\ldots \\}, {\\Bbb R}_+ \\equiv [0,\\infty)$.\n \nWe write $X_n \\mbox{$\\rightarrow_P\\ $} X$ if $X_n$ converges to $X$ in probability, $X_n\n\\Rightarrow X$ if $X_n$ converges to $X$ in distribution, and $X \\stackrel{\\mbox{\\small d}}{=} Y$\nif $X$ and $Y$ have the same distribution.  We use $\\mbox{\\bf 1}$ to denote\nindicator functions, so that $\\mbox{\\bf 1}(A)=1$ if $A$ is true and\n$\\mbox{\\bf 1}(A)=0$ otherwise.\n\n\n\n \n\n\n\n \\section{Independent random variables conditioned on a weighted\nsum}\\label{sect2}\n\n\n\\subsection{The combinatorial setup}\n\nCommon to the enumerations (\\ref{Nassembly}) through (\\ref{Nselection})\n is the form\n\\begin{equation}\\label{Ngeneral}\n N(n, \\b{a})  \\equiv   | \\{  \\b{C} = \\b{a} \\} | = \n \\mbox{\\bf 1}(a_1+2 a_2 + \\cdots + n a_n =n) \n\\  f(n)  \\ \\prod_1^n g_i(a_i),\n\\end{equation}\nwith $f(n)=n!$ for assemblies, and $f(n) \\equiv 1$ for multisets and\nselections. \nTo see that (\\ref{Ngeneral}) involves independent random variables\nconditioned on a weighted sum, view the right hand side as a product of\nthree factors.\nFirst, the  indicator function, which depends\non both $n$ and $\\b{a}$, corresponds to conditioning on the value of a\nweighted sum. Second, the factor $f(n)$ does not depend \non $\\b{a}$, and hence disappears from\nconditional probabilities.  The product form of the third factor \ncorresponds to $n$ mutually independent, but not identically distributed,\nrandom variables. \n\nThe distribution of a random assembly, multiset, or selection $\\b{C}(n)$ \ngiven in (\\ref{combdist.old}) can now be expressed in the following form. \nFor $\\b{a} \\in\n\\B{Z}_+^n$,\n\\begin{equation}\\label{combdist}\n\\B{P}(\\b{C}(n)=\\b{a})  =\n \\mbox{\\bf 1}(a_1+2 a_2 + \\cdots + n a_n =n) \n\\  \\frac{f(n)}{p(n)}  \\ \\prod_1^n g_i(a_i).\n\\end{equation}\n\nGiven functions $g_1,g_2,\\ldots$ from $\\B{Z}_+ $ to $\\B{R}_+$, and a\nconstant $x>0$, let $Z_1,Z_2, \\ldots$ be independent nonnegative\ninteger valued \nrandom variables\nwith distributions given by\n\\begin{equation}\\label{not used 1}\n\\B{P}(Z_i=k) = c_i \\ g_i(k) \\ x^{ik}, \\quad k=0,1,2,\\ldots \\ .\n\\end{equation}\nThe above definition, in which $c_i \\equiv c_i(x)$ is the normalizing\nconstant, makes sense if and only if the value of $x$ and the functions\n$g_i$ are such that\n\\begin{equation}\\label{normconstant}\nc_i \\ \\equiv \\ \\left( \\sum_{k \\ge 0} g_i(k) x^{ik} \\right)^{-1}\n\\ \\ \\in (0,\\infty).\n\\end{equation}\n\n\nFor assemblies, $g_i(k) =(m_i\/i!)^k\/k!$, so that (\\ref{normconstant}) is\nsatisfied for all $x>0$. Defining $\\lambda_i \\equiv m_i \\ x^i\/i!$,\nwe see that $c_i = \\exp(-\\lambda_i)$ and $Z_i$ is Poisson with mean and\nvariance \n\\begin{equation}\\label{Zassembly}\n\\B{E} Z_i = \\mbox{var}(Z_i) = \\lambda_i \\equiv \\frac{m_i x^i}{i!}.\n\\end{equation}\nFor multisets, $g_i(k) = {m_i + k-1 \\choose k}$, so the summability\ncondition (\\ref{normconstant}) is satisfied if and only if $x < 1$.  For\n$x \\in (0,1)$, we have $c_i = (1-x^i)^{m_i}$ and $Z_i$ has the negative\nbinomial distribution with parameters $m_i$ and $x^i$ given by\n\\[\n\\B{P}(Z_i =k) = {m_i +k-1 \\choose k} (1-x^i)^{m_i} \\ x^{ik}, \\ \\ \\\nk=0,1,2,\\ldots,\n\\]\nwith mean and variance \n\\begin{equation}\\label{Zmultiset}\n\\B{E} Z_i = \\frac{m_i x^i}{1-x^i}, \\ \\ \\ \\ \\mbox{var}(Z_i) = \n\\frac{m_i x^i}{(1-x^i)^2}.\n\\end{equation}\nIn the special case $m_i=1$, this is just the geometric distribution,\nand in general $Z_i$ is the sum of $m_i$ independent random variables \neach with the geometric distribution  $\\B{P}(Y=k)=(1-x^i)x^{ik}$ for $k \\ge\n0$.\n\nFor selections, $g_i(k)={m_i \\choose k}$, which is zero for $k > m_i$,\nso that\n(\\ref{normconstant}) is satisfied for all $x>0.$  We see that \n $c_i= (1+x^i)^{-m_i}$, by writing\n\\[\n\\B{P}(Z_i=k)= c_i {m_i \\choose k} x^{ik} ={m_i \\choose k} \\left(\n\\frac{x^i}{1+x^i} \\right)^k \\left( \\frac{1}{1+x^i} \\right)^{m_i-k}.\n\\]\nThus, with $p_i = x^i\/(1+x^i)$, the distribution of $Z_i$ is \nbinomial with parameters $m_i$ and $p_i$,  with mean and variance\n\\begin{equation}\\label{Zselection}\n\\B{E} Z_i = m_i \\ p_i= \\frac{m_i x^i}{1+x^i},\n \\ \\ \\ \\ \\mbox{var}(Z_i)=m_i \\ p_i \\ (1-p_i)= \\frac {m_i x^i}{(1+x^i)^2}.\n\\end{equation}\n\n\n\\subsection{Conditioning on  weighted sums in general}\nIn order to give a proof of (\\ref{equaldist}) which will also serve in\nSection \\ref{sect6} on process refinements, and Section \\ref{sect8} on\nlarge deviations, \nwe generalize to a situation\nthat handles weighted sums with an arbitrary finite index set.  We\nassume that $I$ is a finite set, and for each $\\alpha \\in I,\n g_\\alpha : \\B{Z}_+ \\rightarrow \\B{R}_+$ is given. \nWe assume that $w$ is a given weight function with values in\n $\\B R$ or more generally $\\B{R}^d$, so that for $\\alpha \\in I,\nw(\\alpha)$ is the weight of $\\alpha$.  For the combinatorial examples\nin  Section\n\\ref{sect1}, we had $I=\\{1,2,\\ldots,n \\}$, and a one--dimensional space\nof weights, with $w(i)=i$.  \nFor $\\b{a} \\in \\B{Z}_+^I$ with coordinates $a_\\alpha \\equiv a(\\alpha),$\nwe use vector dot product notation for the weighted sum\n\\[\n\\b{w} \\cdot \\b{a} \\equiv \\sum_{\\alpha \\in I} a(\\alpha) w(\\alpha) .\n\\]\n\nFurthermore, we assume that we are given a\ntarget value $t$ such that there exists a normalizing\nconstant $f(I,t)$ so that the formula\n\\begin{equation}\\label{gencombdist}\n \\B{P}(\\b{C}_I=\\b{a}) = \\mbox{\\bf 1}(\\b{w}\\cdot \\b{a} = t) f(I,t) \\prod_{\\alpha\n\\in I} g_{\\alpha}(a_{\\alpha}), \\quad \\b{a} \\in \\B{Z}_+^I \n\\end{equation}\ndefines a  probability distribution for a stochastic process $\\b{C}_I$ with\nvalues in  $\\B{Z}_+^I$.   \nThe distribution of $\\b{C}(n)$ given by (\\ref{combdist}) is a special\ncase\nof (\\ref{gencombdist}) with $t=n$ and $f(I,t)= f(n)\/p(n).$ \n\nAssume that for some value $x>0$ there exist normalizing constants\n $c_\\alpha \\equiv c_\\alpha(x) \n\\in (0,\\infty)$, such that for each $\\alpha \\in I$,\n\\begin{equation}\n \\ \\B{P}(Z_\\alpha = k) = c_\\alpha(x) g_\\alpha (k) x^{w(\\alpha)k},\n   \\ k=0,1,2,\\ldots \n\\label{genZdist}\n\\end{equation}\ndefines a probability distribution on $\\B{Z}_+$.  In case $d >1$, so\nthat $w(\\alpha)=(w_1(\\alpha),\\ldots,w_d(\\alpha))$, we take $x \\equiv\n(x_1,\\ldots,x_d) \\in (0,\\infty)^d$, and $x^{w(\\alpha)k}$ denotes the\nproduct $x_1^{w_1(\\alpha)k}\\cdots x_d^{w_d(\\alpha)k}$.\n Define the weighted sum\n$T$\nby\n\\begin{equation}\n\\label{Tdef}\nT \\equiv T_I \\equiv \\sum_{\\alpha \\in I} w(\\alpha) Z_\\alpha.\n\\end{equation}\n\nIt should now be clear that the following is a generalization of\n(\\ref{equaldist}).\n\\begin{theorem}\\label{genequaldist}\nLet $\\b{Z}_I \\equiv (Z_\\alpha)_{\\alpha \\in I}$ have independent \ncoordinates $Z_\\alpha$\nwith distribution given by (\\ref{genZdist}), and let $\\b{C}_I$ have the \ndistribution given by (\\ref{gencombdist}). Then\n\\begin{equation}\n\\label{equation genequaldist}\n\\b{C}_I \\stackrel{\\mbox{\\small d}}{=} \\left( \\b{Z}_I \\left| \\right. T=t \\right),\n\\end{equation}\nand hence for any $B \\subset I$, the processes restricted to indices in\n$B$ satisfy\n\\begin{equation}\n\\label{equation genequaldist1}\n\\b{C}_B \\stackrel{\\mbox{\\small d}}{=} \\left( \\b{Z}_B \\left| \\right. T=t \\right).\n\\end{equation}\nFurthermore, the normalizing constants and the conditioning probability are \nrelated by \n\\begin{equation}\n\\label{probT=t}\n\\B{P}(T=t) = f(I,t)^{-1} \\  x^t \\prod_{\\alpha \\in I} c_\\alpha(x).\n\\end{equation}\n\\end{theorem}\n\n\\noindent{\\bf Remark:} \nThe distribution of $\\b Z_I$, and hence that of $T \\equiv \\b\nw \\cdot \\b Z_I$, depends on $x$, so the left side $\\B P(T=t)$ of\n(\\ref{probT=t}) is a function of $x$.\n\n\\noindent{\\bf Proof\\ \\ } The distribution of $\\b{Z}_I$ is given by\n\\[\n \\B{P}(\\b{Z}_I=\\b{a})= \n \\prod_{\\alpha \\in I} \\left( c_\\alpha g_\\alpha (a_\\alpha)\n x^{w(\\alpha)a(\\alpha)} \\right) =\nx^{\\b{w} \\cdot \\b{a}} \\ \\ \\prod_{\\alpha \\in I} c_\\alpha \\ \\\n \\prod_{\\alpha \\in I} g_\\alpha (a_\\alpha),\n\\] \nfor  $\\b{a} \\in \\B{Z}_+^I$, so that\nif $\\b{w} \\cdot \\b{a} = t$  then \n\\begin{equation}\\label{new1}\n\\B{P}(\\b{Z}_I=\\b{a}) = x^t \\ \\prod c_\\alpha \\ \nf(I,t)^{-1} \\  \\B{P}(\\b{C}_I=\\b{a}).\n\\end{equation}\nThe conditional distribution of  $\\b{Z}_I$ given $\\{ T=t \\}$ is \ngiven by\n\\begin{eqnarray}\n\\B{P}(\\b{Z}_I=\\b{a} | T=t) & = &\n \\frac{\\mbox{\\bf 1}(t= \\b w \\cdot \\b a ) \\B{P}(\\b{Z}_I=\\b{a})}  \n   { \\B{P}(T=t)}    \\nonumber \\\\\n&&\\nonumber \\\\ & = &\n \\frac{x^t \\ (\\prod c_\\alpha) f(I,t)^{-1} \\B{P}(\\b{C}_I=\\b{a})}  \n   { \\B{P}(T=t)}\n\\label{step1} \\\\\n&&\\nonumber \\\\& = &\n \\frac{\\ \\ \\ \\ x^t \\ (\\prod c_\\alpha) f(I,t)^{-1} \\B{P}(\\b{C}_I=\\b{a})}  \n   { \\sum_{\\b{b} \\in \\B{Z}_+^I}x^t \\ (\\prod c_\\alpha) \\ \nf(I,t)^{-1}\n \\ \\B{P}(\\b{C}_I=\\b{b})}\n\\nonumber\\\\\n&&\\nonumber \\\\& = &\n\\frac{ \\B{P}(\\b{C}_I=\\b{a})}  \n   { \\sum_{\\b{b} } \\B{P}(\\b{C}_I=\\b{b})} \\nonumber \\\\\n&&\\nonumber \\\\\n& = & \\B{P}(\\b{C}_I=\\b{a}), \\ \\b{a} \\in \\B{Z}_+^I. \n\\label{step2}\n\\end{eqnarray}\nThe equality between (\\ref{step1}) and (\\ref{step2}), for any $\\b{a}$ for\nwhich $\\B{P}(\\b{C}_I=\\b{a})>0$, establishes (\\ref{probT=t}).\\hfill \\mbox{\\rule{0.5em}{0.5em}}\n\nFor the combinatorial objects in Section {\\ref{sect1}, $I=\\{\n1,2,\\ldots,n \\}$, and $w(i)=i$.  For this case $T$ reduces to\n\\begin{equation}\n    T \\equiv T_n \\equiv Z_1 + 2 Z_2 + \\cdots + n Z_n.\n\\label{deftn}\n\\end{equation}\nIn the  case of assemblies, corresponding to (\\ref{Nassembly}) and\n(\\ref{Zassembly}),  the distribution of $Z_i$ is Poisson $(\\lambda_i)$, \nand (\\ref{probT=t}) reduces to\n\\begin{equation}\n\\label{assprobT=t}\n\\B P (T_n=n) \\  = \\ \\frac{p(n)}{n!} \\ x^n \\ \\exp(-\\lambda_1-\\cdots\n-\\lambda_n),\n\\end{equation}\nwhere  $\\lambda_i = m_i x^i \/i!$ and $ x>0$.\nIn the case of multisets, corresponding to (\\ref{Nmultiset})\nand (\\ref{Zmultiset}),  $Z_i$ is distributed like \nthe sum of $m_i$ independent\ngeometric $(x^i)$ random variables, and (\\ref{probT=t}) reduces to\n\\begin{equation}\n\\label{multiprobT=t}\n\\B P (T_n=n)= \\ p(n) \\ x^n \\ \\prod_1^n (1-x^i)^{m_i},\n\\end{equation}\nfor $0<x<1$.\nIn the case of selections, corresponding to (\\ref{Nselection}) and\n(\\ref{Zselection}), the distribution of  $Z_i$\nis binomial $(m_i, x^i\/(1+x^i))$, so that (\\ref{probT=t}) reduces to\n\\begin{equation}\n\\label{selectprobT=t}\n\\B P (T_n=n) = p(n) \\ x^n \\ \\prod_1^n (1+x^i)^{-m_i},\n\\end{equation}\nfor $x>0$.\n\n\n\n \n\n\n \n\\section{Total variation distance}\\label{sect3}\n\n\nA useful way to establish that the independent process $\\b Z_n \\equiv\n(Z_1,Z_2,\\ldots,Z_n)$ is a good approximation for the dependent\ncombinatorial process $\\b C(n)$ is to focus on a subset $B$ of the\npossible component sizes, and give an upper bound on the total variation\ndistance between the two processes, both restricted to $B$.  Theorem\n\\ref{tvthm} below shows how this total variation distance for\nthese two processes reduces to the total variation distance between \ntwo one--dimensional random variables.  \n\nHere is a quick review of the relevant features of total variation\ndistance. For two random elements $X$ and $Y$ of a finite or countable\nspace $S$,\nthe total variation distance between $X$ and $Y$ is defined by\n\\[\nd_{TV}(X,Y) = \\frac{1}{2} \\sum_{s \\in S} |\\B P(X=s) - \\B P(Y=s)|.\n\\]\nProperly speaking this should be referred to as the distance between the\ndistribution ${\\mathcal L}(X)$ of $X$ and the distribution ${\\mathcal L}(Y)$ of\n$Y$, written for example as $d_{TV}({\\mathcal L}(X),{\\mathcal L}(Y))$.\nThroughout this paper we use the simpler notation, except in Section\n\\ref{sect8} which involves changes of measure.\n\nMany authors, following the tradition of analysis of signed measures,\n omit the factor of $1\/2$.  Using the factor of\n$1\/2$, we have that $d_{TV}(X,Y) \\in [0,1]$, and furthermore, $d_{TV}$ is\nidentical to the Prohorov metric, providing the underlying metric on $S$\nassigns distance $\\ge 1$ between any two distinct points. \nIn particular,\na sequence of random elements $X_n$ in a discrete space $S$ converges in\ndistribution to $X$ if and only if $d_{TV}(X_n,X) \\rightarrow 0$. \n \nAnother characterization of total variation distance\n is\n\\[\nd_{TV}(X,Y) = \\max_{A \\subset S}\\left( \\B P(X \\in A) - \\B P(Y \\in A)\n\\right),\n\\]\nand in the discrete case, \na necessary and sufficient condition that the maximum be achieved by $A$\nis that $\\{s: \\B P(X=s) > \\B\nP(Y=s) \\}  \\subset A \\subset \\{s: \\B P (X=s) \\ge \\B P(Y=s) \\}$.\n\nThe most intuitive description of total variation distance is in terms\nof coupling.  A\n``coupling'' of $X$ and $Y$ is a probability measure on $S^2$ whose\nfirst and second marginals are the distributions of $X$ and $Y$\nrespectively.  Less formally, a coupling of $X$ and $Y$ is a \nrecipe for constructing $X$ and $Y$ simultaneously on the same\nprobability space, subject only to having given marginal distributions for\n$X$ and for $Y$. In terms of all possible coupling measures on $S^2$,\n\\begin{equation}\\label{tvcouple}\nd_{TV}(X,Y) = \\min_{couplings} \\B P (X \\ne Y).\n\\end{equation}\nThe minimum above is achieved, but\n in general there is not a unique optimal coupling. In fact\na discrete coupling achieves $\\B P (X \\ne Y) = d_{TV}(X,Y)$, if and only if, for\nall $s \\in S, \\ \\B P(X=Y=s) = \\min ( \\B P(X=s),\\B P (Y=s))$. \nIntuitively, if  $d_{TV}(X,Y)$  is small, then $X$ and $Y$ are nearly\nindistinguishable from a single observation; formally, for any\nstatistical test to decide whether $X$ or $Y$ is being observed, the sum\nof the type I and type II errors is at least $1-$  $d_{TV}(X,Y)$ .\n\nUpper bounds on the total variation distance between a combinatorial\nprocess and a simpler process are useful because these upper bounds are\ninherited by functionals of the processes.  If  $h:S\n\\rightarrow T$ is a deterministic map between countable spaces,\n and $X$ and $Y$ are random\nelements of $S$, so that $h(X)$ and\n$h(Y)$ are random elements of $T$, then\n\\begin{equation}\nd_{TV}(h(X),h(Y)) \\ \\leq \\ d_{TV}(X,Y).\n\\label{tvinequality}\n\\end{equation}\nTheorem \\ref{tvthm} below, and its refinement, Theorem \\ref{tvrefine} in\nSection \\ref{sect6}, both describe combinatorially interesting cases in\nwhich  equality holds in (\\ref{tvinequality}).  It is natural to\nask when, in general, such equality holds. The following elementary\ntheorem provides an answer. \n\n\\begin{theorem}\\label{tv=thm}\nIn the discrete case, equality holds in (\\ref{tvinequality}) if and only\nif the sign of $\\B P(X=s)- \\B P(Y=s)$ depends only  on $h(s)$, in the\nnon-strict sense that  \n $\\forall a,b \\in S$,\n\\[ h(a)=h(b) \\mbox{ implies } \n\\left(\\B P(X=a) - \\B P(Y=a)\\right)\\ \\left(\\B P(X=b)-\\B P(Y=b)\\right) \\ \\ \\ge 0.\n\\]\n\\end{theorem}\n\\noindent{\\bf Proof\\ \\ }\nConsider the proof of (\\ref{tvinequality}), namely\n\\begin{eqnarray}\n2 d_{TV}(h(X),h(Y))& =& \\sum_{r \\in T} \\left| \\B P(h(X)=r)-\\B P(h(Y)=r)\n\\right| \\nonumber \\\\\n&&\\nonumber \\\\ \n&=& \\sum_r \\left| \\sum_{a \\in S: h(a)=r} (\\B P(X=a)-\\B P(Y=a))\n\\right| \\label{cancelstep} \\\\\n&& \\nonumber \n\\\\ &\\leq & \\sum_r  \\sum_{a: h(a)=r}\\left| \\B P(X=a)-\\B P(Y=a)\n\\right| \\label{canstep} \\\\\n&& \\nonumber\\\\ \n&=& 2 d_{TV}(X,Y). \\nonumber  \n\\end{eqnarray}\nSince the inequality in (\\ref{canstep}) holds term by term in the outer\nsums,\nequality holds overall if and only if equality holds for each $r$. This\n in turn is equivalent to the condition that   for each $r$,\n there are no terms of opposite sign in the inner sum\nin (\\ref{cancelstep}).\\hfill \\mbox{\\rule{0.5em}{0.5em}}\n\nDiaconis and Pitman (1986) view ``sufficiency'' as a\nkey concept.  In the context above, $h: S \\rightarrow T$ is a sufficient\nstatistic for discriminating between the distributions of $X$ and $Y$ in\n$S$, if the likelihood ratio depends only on $h$; i.e. if there is a\nfunction $f: T \\rightarrow \\B R$ such that for all $s \\in S, \\ \\B P(X=s)\n= f(h(s)) \\ \\B P(Y=s)$. Taking a sufficient statistic preserves total\nvariation distance, as observed by\n Stam (1978). This is also a special case of Theorem \n\\ref{tv=thm}, in\nwhich a product is nonnegative because it is a square: $ (\\B P(X=a)-\\B\nP(Y=a))(\\B P(X=b)-\\B P(Y=b)) \\ = \\ (f(h(a))-1)(f(h(b))-1)\\B P(Y=a) \\B\nP(Y=b) \\geq 0$ whenever $h(a)=h(b)$.\n \n\\begin{theorem}\\label{tvthm}\nLet $I$ be a finite set, and for $\\alpha \\in I$, let $C_\\alpha$ and\n$Z_\\alpha$ be $\\B Z_+$ valued random variables, such that the\n$Z_\\alpha$ are mutually independent.  Let $\\b w = (w(\\alpha))_{\\alpha \\in\nI}$ be a deterministic weight function on $I$ with values in some linear\nspace, let $T = \\sum_{\\alpha \\in\nI} w(\\alpha) Z_\\alpha$, and let $t$ be such that\n$\\ \\B P(T=t) >0$.  For $B \\subset I$, \nwe use the notation $\\b C_B \\equiv (C_\\alpha)_{\\alpha \\in\nB}$ and $\\b Z_B \\equiv (Z_\\alpha)_{\\alpha \\in B}$ for\nrandom elements of $\\B Z_+^B$.  Define \n\\[\nR \\equiv R_B \\equiv \\sum_{\\alpha \\in B} w(\\alpha) Z_\\alpha,\\ \\  \nS \\equiv S_B \\equiv \\sum_{\\alpha \\in I-B} w(\\alpha) Z_\\alpha,\n\\]\nso that $T = R+S$ and $R$ and $S$ are independent.\nIf \n\\begin{equation}\n\\b C_I \\stackrel{\\mbox{\\small d}}{=} (\\b Z_I | T=t),\n\\label{tvhyp}\n\\end{equation} \n then\n\\begin{equation}\\label{tvhyp1}\nd_{TV}(\\b C_B, \\b Z_B) \\ = d_{TV}( (R_B|T=t), R_B).\n\\end{equation}\n\\end{theorem}\n\\noindent{\\bf Proof\\ \\ }\nWe present two proofs, since it is instructive to contrast them.\n\\def\\b w \\cdot \\b a{\\b w \\cdot \\b a}\nNote that not only are $R$ and $S$ independent, but also that $R$ is a\nfunction of $\\b Z_B$, and $\\b Z_B$ and $S$ are independent. \n For $\\b a \\in \\B Z_+^B$, write $\\b w \\cdot \\b a \\equiv \\sum_{\\alpha \\in B}\nw(\\alpha) a(\\alpha)$. \n\\def\\frac{1}{2}{\\frac{1}{2}}\n\\begin{eqnarray*}\nd_{TV}(\\b C_B, \\b Z_B) & = & \\frac{1}{2} \\sum_{\\b a \\in \\B Z_+^B}\n \\left| \\B P( \\b Z_B = \\b a \\left| T=t \\right. ) - \n\\B P( \\b Z_B = \\b a) \\right| \\\\\n& = &\n   \\frac{1}{2} \\sum_r \\ \\ \\ \\sum_{\\b a: \\b w \\cdot \\b a = r}\n \\left| \\frac{\\B P( \\b Z_B = \\b a, r+S=t)}{\\B P(T=t)}\n   - \\B P( \\b Z_B =\\b a) \\right|\n\\\\ && \\\\ & = &\n   \\frac{1}{2} \\sum_r \\ \\ \\ \\sum_{\\b a: \\b w \\cdot \\b a = r}\n \\left| \\frac{\\B P( \\b Z_B = \\b a) \\B P( r+S=t)}{\\B P(T=t)}\n   - \\B P( \\b Z_B =\\b a) \\right|\n\\\\ && \\\\ & = &\n   \\frac{1}{2} \\sum_r \n \\left| \\frac{\\B P( R=r) \\B P( r+S=t)}{\\B P(T=t)}\n   - \\B P( R=r ) \\right|\n\\\\ && \\\\ & = &\n   \\frac{1}{2} \\sum_r \\left| \\frac{\\B P(R=r,r+S=t)}{\\B P(T=t)}\n   - \\B P(R=r) \\right|\n\\\\ && \\\\ &=&\n   \\frac{1}{2} \\sum_r \\left| \\B P(R=r|T=t) - \\B P(R=r) \\right|\n\\\\ && \\\\ &=& d_{TV}( (R|T=t), R).\n\\end{eqnarray*}\n\nHere is a second proof of  Theorem \\ref{tvthm}, viewed \nas a corollary of Theorem \\ref{tv=thm}, with the functional $h$ on $\\B\nZ_+^B$ defined by $h(\\b a) = \\b w \\cdot \\b a$.  We need only observe that $h$ is a\nsufficient statistic since $\\B P(\\b Z_B=\\b a|T=t)= {\\Bbb P}(\\b Z_B=\\b a)\n{\\Bbb P}(S=t-h(\\b a))\/{\\Bbb P}(T=t) \\ $.   \\hfill \\mbox{\\rule{0.5em}{0.5em}} \n\n\nFor the sake of calculations of total variation distance between a\ncombinatorial process and its independent process approximation, the\nmost useful form for the conclusion of Theorem \\ref{tvthm} is\n\\begin{eqnarray}\nd_{TV}(\\b C_B, \\b Z_B) & = &\n   \\frac{1}{2} \\sum_r \n \\left| \\frac{\\B P( R=r) \\B P( r+S=t)}{\\B P(T=t)}\n   - \\B P( R=r ) \\right| \\nonumber\n\\\\ && \\nonumber \\\\ & = &\n   \\frac{1}{2} \\sum_r \\B P( R=r) \n \\left| \\frac{\\B P( S=t-r)}{\\B P(T=t)}\n   - 1 \\right|. \\label{tvcalc}\n\\end{eqnarray}\nIn the  usual combinatorial case, where $t=n$ and\n$T=Z_1+2Z_2+\\cdots+nZ_n$, this gives \n\\begin{equation}\nd_{TV}(\\b C_B, \\b Z_B)=\n \\frac{1}{2} \\B P(R>n) +\\frac{1}{2} \\sum_{r=0}^n \\B P( R=r) \n \\left| \\frac{\\B P( S=n-r)}{\\B P(T=n)}\n   - 1 \\right|. \\label{tvcalc1}\n \\end{equation}\n    \n\nThere are two elementary observations that point to strategies for giving\nupper bounds on total variation distance.  First, for discrete random\nelements we have in general\n\\begin{eqnarray*}\nd_{TV}(X,Y) & \\equiv & \\frac{1}{2} \\sum_{s \\in S} |\\B P(X=s) - \\B P(Y=s)|\n\\\\ && \\\\\n&=& \\sum_{s \\in S} \\left( \\B P(X=s) - \\B P(Y=s) \\right)^+\n\\\\ && \\\\\n&=& \\sum_{s \\in S} \\left( \\B P(X=s) - \\B P(Y=s) \\right)^-,\n\\end{eqnarray*}\nwhere the notation for positive and negative parts is such that, for\nreal $x$, $x= x^+ - x^-,$ and $|x|=x^+ + x^-$.  In the context of \n(\\ref{tvcalc}) this is\nuseful in the following form. Let $A \\subseteq I$. Then\n\\begin{eqnarray}\nd_{TV}(\\b C_B, \\b Z_B) & = &\n   \\sum_r \\B P( R=r) \n \\left(1 - \\frac{\\B P( S=t-r)}{\\B P(T=t)}\n    \\right)^+ \\nonumber \\\\ && \\nonumber \\\\\n& \\leq & \\B P(R \\not \\in A) \\ \\ + \\sup_{r \\in A} \n      \\left(1 - \\frac{\\B P( S=t-r)}{\\B P(T=t)}\n    \\right)^+. \\label{truncstrategy} \n\\end{eqnarray}\nSpecializing to the case where the weighted sum $R$ is real\nvalued, and $A = \\{0,1,2,\\ldots,k\\}$, the truncation level $k$ \nis chosen much larger than $\\B E R$,\nso that large deviation theory can be used to bound $\\B P(R > k)$, but\nnot too large, so that $\\B P( S=t-r)\/ \\B P(T=t)$ can be controlled to\nshow it is close to one.  \n\nThe second elementary observation, which is proved and exploited in\n{\\rm Arratia and Tavar\\'e (1992a)}, is that the denominator in (\\ref{tvcalc}) can be replaced by any\nconstant $c>0$, at the price of at most a factor of 2, in the sense that\nfor independent $R$ and $S$ such that $\\B P(R+S=t) >0$,\n\\[\n\\frac{1}{2} \\sum_r \\B P(R=r) \\left|\\frac{\\B P(S=t-r)}{\\B P(R+S=t)} -1\n\\right| \n\\leq  \n \\sum_r \\B P(R=r) \\left|\\frac{\\B P(S=t-r)}{c} -1 \\right| .\n\\]\nBy using this, for example with $c=\\B P(S=t)$, \ngiving an upper bound on the total variation distance for\ncombinatorial process approximations is reduced to showing that the\ndensity of $S$ is relatively constant.\n\nLower bounds for variation distance are often more difficult to obtain,\nbut it is worth noting that in the combinatorial setup, \nsince $\\{R_B>n\\} \\subseteq \\{\\mbox{\\boldmath $C$}_B \\neq\n\\mbox{\\boldmath $Z$}_B\\}$, we have, without the factor $\\frac{1}{2}$ suggested by\n(\\ref{tvcalc1}),\n\\begin{equation}\\label{tvlower}\nd_{TV}(\\b C_B,\\b Z_B) \\ge {\\Bbb P}(R_B >n).\n\\end{equation}\n\n \n\n\n  \\section{Heuristics for useful approximation}\\label{sect4}\n \n Recall first that for $B \\subset [n]$, we have  \n $\\b C_B \\stackrel{\\mbox{\\small d}}{=} (\\b Z_B | T_n = n)$.\n If $d_{TV}(\\b C_B, \\b Z_B)$ is small, the approximation of $\\b C_B$ by\n $\\b Z_B$ is useful. Probabilistic intuition suggests that conditioning\n on $T_n = n$ does not change the distribution of $\\b Z_B$ by much,\n provided that the event $\\{T_n = n\\}$ is relatively likely. This in turn\n corresponds to a choice of $x = x(n)$ for which ${\\Bbb E} T_n$ is\n approximately $n$. Let $\\sigma_n^2 \\equiv {\\rm var}(T_n),$\n and  let $\\sigma_B^2 = {\\rm var}(R_B)$.\n Intuition then suggests that {\\em if}\n \\begin{equation}\\label{metathm1}\n \\frac{n - {\\Bbb E}(T_n)}{\\sigma_n} \\mbox{\\it\\ is not large}\n\\end{equation}\n{\\em and}\n\\begin{equation}\\label{metathm1a}\n  \\frac{{\\Bbb E} R_B}{\\sigma_n}\n \\mbox{\\it\\ and }\\frac{\\sigma_B}{\\sigma_n} \\mbox{\\it\\ are small}\n \\end{equation}\n {\\em then} $d_{TV}(\\b C_B, \\b Z_B)$ {\\em is small.}\n\n While our main focus is on the appropriate choice of $x$, we also\n discuss below the appropriate choice of $B$ for examples including\n permutations, mappings, graphs, partitions of sets, and partitions of\n integers.\n \n There is an important qualitative distinction between cases in which the\n appropriate $x$ is constant, and those in which $x$ varies with $n$.\n If $x$ does not depend on $n$, then a single independent process $\\b Z =\n (Z_1,Z_2,\\ldots)$ may be used to approximate $\\b C(n) \\equiv\n (C_1(n),\\ldots,C_n(n))$, which we identify with\n $(C_1(n),\\ldots,C_n(n),0,0,\\ldots) \\in {\\Bbb Z}_+^\\infty$.\n Under the usual product topology on ${\\Bbb Z}_+^\\infty,$ we have that \n  $\\b C(n) \\Rightarrow \\b Z$  if, and only if,  for\n every fixed $b$, $\\b C_b(n) \\equiv\n (C_1(n),\\ldots,C_b(n)) \\Rightarrow \\b Z_b \\equiv (Z_1,\\ldots,Z_b)$\n as random elements in $\\B Z_+^b$.\n  Since the metric on ${\\Bbb Z}_+^b$ is discrete, we conclude\n that $\\b C_b(n) \\Rightarrow \\b Z_b$ if, and only if, for each fixed $b$, \n $d_{TV}(\\b C_b(n),\\b Z_b) \\to 0$. For cases where $x$, and hence $\\b Z$, varies\n with $n$, it makes no sense to write \n $\\b C(n) \\Rightarrow \\b Z$.\n However, it is still useful to be able to\n estimate $d_{TV}(\\b C_B(n),\\b Z_B(n))$.\n \n We discuss first considerations involved in the choice of $x$ and $B$, and then\n heuristics for predicting the accuracy of approximation.\n \n \\subsection{Choosing the free parameter $x$}\n \n It is convenient to discuss the three basic types of combinatorial\n structure separately.\n \n \\subsubsection{Assemblies}\n It follows from (\\ref{Zassembly}) that\n \\begin{equation}\\label{assem-etn}\n {\\Bbb E} T_n \\equiv \\sum_{i=1}^n i {\\Bbb E} Z_i = \\sum_{i=1}^n \\frac{m_i x^i}{(i-\n 1)!},\n \\end{equation}\n while \n \\begin{equation}\\label{assem-vartn}\n \\sigma_n^2 = \\sum_{i=1}^n i^2 {\\Bbb E} Z_i = \\sum_{i=1}^n \\frac{i^2 m_i x^i}{i!}.\n \\end{equation}\n In the case of permutations, we take $x = 1$ to see that ${\\Bbb E} T_n = n$,\n and $\\sigma_n^2 = n(n+1)\/2$. In {\\rm Arratia and Tavar\\'e (1992a)}\\ it is proved that $d_{TV}(\\b C_B,\\b\n Z_B) \\rightarrow 0$ as $n\\rightarrow \\infty,$ with $B=B(n)$, if and\n only if $|B|=o(n)$.  \n\nFor the class of assemblies which satisfy the additional condition\n \\begin{equation}\\label{assembly-hyp}\n \\frac{m_i}{i!} \\sim \\frac{\\kappa y^i}{i} {\\rm\\ as\\ }i \\to \\infty,\n \\end{equation}\n where $y > 0$ and $\\kappa > 0$ are constants, we see that\n$$\n\\frac{{\\Bbb E} T_n}{n} \\to \\left\\{\n\\begin{array}{cl}\n  0, & {\\rm \\ if\\ } 0 < x < y^{-1} \\\\\n  \\kappa, & {\\rm \\ if\\ } x = y^{-1}\\\\\n \\infty, & {\\rm \\ if\\ } x > y^{-1}.\n\\end{array}\n\\right.\n$$\n Hence the only fixed $x$ that ensures that ${\\Bbb E} T_n \\bothsides n$ is\n $x = y^{-1}$, in which case \n \\begin{equation}\\label{mean-varass}\n {\\Bbb E} T_n \\sim n \\kappa, \\  \\sigma_n \\sim n\n \\sqrt{\\frac{\\kappa}{2}}.\n \\end{equation} \n \n For the example of random mappings,\n $$\n m_i = e^i (i-1)!\\, {\\Bbb P}({\\rm Po}(i) < i),\n $$\n where ${\\rm Po}(i)$ denotes a Poisson random variable with mean $i$;\nsee Harris (1960), Stepanov (1969).\n It follows that we must take $x = 1\/e$, and, from the Central Limit\n Theorem, $\\kappa = 1\/2$. \n In this case ${\\Bbb E} T_n \\sim n\/2$ and $ \\sigma_n \\sim n\/2$.\n \n For the example of random graphs, with all $2^{{n \\choose 2}}$ graphs\n equally likely, the fact that the probability of being connected tends\n to 1 means that the constant vector $(0,0,\\ldots,0,1) \\in \\B Z_+^n$ is a\n good approximation, in total variation distance, to $\\b C(n)$.  This is\n a situation in which the equality $\\b C(n) \\stackrel{\\mbox{\\small d}}{=} (\\b Z_n|T_n=n)$\n yields no useful approximation.  With $x$ chosen so that $\\B ET_n=n$,\n and $B=\\{1,2,\\ldots,n-1\\}$, we have that $d_{TV}(\\b C_B,\\b Z_B)\n \\rightarrow 0$, but only because both distributions are close to that\nof the process that is identically 0 on $\\B Z_+^B$.\n \n For partitions of a set, which is discussed further in Section\n \\ref{sect5.2} and Section \\ref{sect10}, with $x=x(n)$ being the solution of\n $xe^x=n$, and $B=\\{1,2,\\ldots,b \\} \\cup \\{c,c+1,\\ldots,n \\}$\n where $b \\equiv b(n)$ and $c \\equiv c(n)$,\n the heuristic (\\ref{metathm1}) suggests that $d_{TV}(\\b C_B, \\b Z_B)\n \\rightarrow 0$ if and only if both $(x-b)\/\\sqrt{\\log n} \\rightarrow\n \\infty$ and $(c-x)\/\\sqrt{\\log n}\n \\rightarrow \\infty$.\n For $B$ of the complementary form\n $B=\\{b,b+1,\\ldots,c \\}$ with $b<c$ both within a bounded number of\n $\\sqrt{\\log n}$ of $x$,  the heuristic suggests that $d_{TV}(\\b C_B, \\b Z_B)\n \\rightarrow 0$ if, and only if, $(c-b)=o(\\sqrt{\\log n})$. Sachkov\n(1974) and Fristedt (1992) have partial results in this area.\n \n \\subsubsection{Multisets}\n Using (\\ref{Zmultiset}) we see that\n \\begin{equation}\\label{multi-etn}\n {\\Bbb E} T_n  = \\sum_{i=1}^n \\frac{i m_i x^i}{1-x^i},\n \\end{equation}\n while \n \\begin{equation}\\label{multi-vartn}\n \\sigma_n^2 = \\sum_{i=1}^n \\frac{i^2 m_i x^i}{(1-x^i)^2}.\n \\end{equation}\n If the multiset construction satisfies the additional hypothesis that\n \\begin{equation}\\label{multiset-hyp}\n m_i \\sim \\frac{\\kappa y^i}{i}\\ {\\rm \\ as\\ } i \\to \\infty,\n \\end{equation}\n where $y > 1$ and $\\kappa > 0$ is fixed, a similar analysis shows\n that the only fixed $x$ that ensures that ${\\Bbb E} T_n \\bothsides n$ is\n $x = y^{-1}$, in which case the asymptotics for ${\\Bbb E} T_n$ and\n$\\sigma_n$ are the same as those in (\\ref{mean-varass}).\n \n The first example that satisfies the hypothesis in (\\ref{multiset-hyp})\n is the multiset in which $p(n) = q^n$ for some integer $q \\geq 2$. In this case\n the $m_i$ satisfy\n \\begin{equation}\\label{poly1}\n q^n = \\sum_{j|n} j m_j,\n \\end{equation}\n so that by the M\\\"obius inversion formula we have\n \\begin{equation}\\label{poly2}\n m_n = \\frac{1}{n} \\sum_{k|n} \\mu(n\/k) q^k,\n \\end{equation}\n where $\\mu(\\cdot)$ is the M\\\"obius function, defined by\n \\begin{eqnarray*}\n \\mu(n) & = & (-1)^k {\\rm\\ if\\ } n {\\rm\\ is\\ the\\ product\\ of\\ } k {\\rm \n \\ distinct\\ primes}\\\\\n \\mu(n) & = & 0 {\\rm\\ otherwise}.\n \\end{eqnarray*}\n It follows from (\\ref{poly1}) that\n $$\n q^i - \\frac{q}{q-1} q^{i\/2} \\leq i m_i \\leq q^i,\n $$\n so that (\\ref{multiset-hyp}) holds with $\\kappa = 1, y = q$. This\n construction arises in the study of necklaces (see Metropolis and Rota\n (1983, 1984) for example), in card shuffling (Diaconis, McGrath and\n Pitman (1994)), and, for $q$ a prime power, in \n factoring polynomials over $GF(q)$, a finite field of $q$ elements. \n In this last case $m_i$ is  the number of irreducible monic polynomials over\n $GF(q)$; see Lidl and Niederreiter (1986), for example. \n \n Another example concerns random mapping patterns. Let $t_n$ denote the number of rooted trees with $n$ unlabelled points, and set $T(x) = \\sum_{n=1}^\\infty t_n x^n.$ Otter (1948) showed that $T(x)$ has radius of convergence $\\rho = 0.3383\\ldots$, from which Meir and Moon (1984) established that\n $$\n m_i \\sim \\frac{\\rho^{-i}}{2i}.\n $$\n  Hence (\\ref{multiset-hyp}) applies with $\\kappa = 1\/2, y = \\rho^{-1}$.\n \n For an example in which $x$ varies with $n$, we consider random\n partitions of the integer $n$. In this case $m_i \\equiv 1$. Taking \n $x = e^{-c\/\\sqrt{n}}$\n and using (\\ref{multi-etn}), we see that\n \\begin{eqnarray*}\n n^{-1} {\\Bbb E} T_n & = & \\sum_{i=1}^n \\frac{n^{-1\/2} i \\exp(-i\n c\/\\sqrt{n})}{1 - \\exp(-i c \/ \\sqrt{n})} \\frac{1}{\\sqrt{n}} \\\\\n & \\to & \\int_0^{\\infty} \\frac{y e^{-c y}}{1 - e^{- c y}} dy \\\\\n & = & \\frac{1}{c^2} \\int_0^1 \\frac{- \\log(1-v)}{v} dv \\\\\n & = & \\frac{\\pi^2}{6 c^2}.\n \\end{eqnarray*}\n Hence to satisfy ${\\Bbb E} T_n \\sim n$, we choose $c = \\pi\/\\sqrt{6}$, so that\n \\begin{equation}\\label{choosex-multi}\n x = \\exp(-\\pi\/\\sqrt{6 n}).\n \\end{equation}\n From (\\ref{multi-vartn}), it follows by a similar calculation that\n \\begin{eqnarray*}\n n^{-3\/2}\\, \\sigma_n^2 & \\to & \\int_0^{\\infty} \\frac{y^2 e^{-c y}}\n {(1 - e^{- c y})^2} dy \\\\\n & = & \\frac{1}{c^3} \\int_0^1 \\left(\\frac{- \\log(1-v)}{v}\\right)^2 dv \\\\\n & = & \\frac{2}{c},\n \\end{eqnarray*}\n so that\n \\begin{equation}\\label{vartn-multipart}\n \\sigma_n^2 \\sim \\frac{2 \\sqrt{6}}{\\pi} n^{3\/2}.\n \\end{equation}\n For sets of the form $B=\\{1,2,\\ldots,b \\} \\cup \\{c,c+1,\\ldots,n \\}$\n where $0 \\leq b \\equiv b(n)$ and $c \\equiv c(n) \\leq n$,\n the heuristic in (\\ref{metathm1}) and (\\ref{metathm1a}) suggests \nthat $d_{TV}(\\b C_B, \\b Z_B)\n \\rightarrow 0$ if, and only if, both $b=o(\\sqrt{n})$ and $c\/\\sqrt{n}\n \\rightarrow \\infty$. For $B$ of the complementary form\n $B=\\{b,b+1,\\ldots,c \\}$ with $b<c$ both of the order of $\\sqrt{n}$,\n   the heuristic suggests that $d_{TV}(\\b C_B, \\b Z_B)\n \\rightarrow 0$ if, and only if, $(c-b)=o(\\sqrt{n})$. See Fristedt\n(1993) and Goh and Schmutz (1993) for related results.\n \n \\subsubsection{Selections}\n In this case, it follows from (\\ref{Zselection}) that\n \\begin{equation}\\label{select-etn}\n {\\Bbb E} T_n  = \\sum_{i=1}^n \\frac{i m_i x^i}{1+x^i},\n \\end{equation}\n while \n \\begin{equation}\\label{select-vartn}\n \\sigma_n^2 = \\sum_{i=1}^n \\frac{i^2 m_i x^i}{(1+x^i)^2}.\n \\end{equation}\n If the selection construction satisfies the additional hypothesis\n (\\ref{multiset-hyp}),\n then, just as for the assembly and multiset constructions, we take \n $x = y^{-1}$, and (\\ref{mean-varass}) holds once more.\n As an example, for square--free factorizations of polynomials over\n a finite field with $q$ elements, we have $y = q, \\kappa = 1, x =\n q^{-1}$.\n \n For an example in which $x$ varies with $n$, we consider once more random\n partitions of the integer $n$ with all parts distinct,\n which is the selection construction with $m_i \\equiv 1$. Taking \n $x = e^{-d\/\\sqrt{n}}$,\n and using (\\ref{select-etn}) we see that\n \\begin{eqnarray*}\n n^{-1} {\\Bbb E} T_n & = & \\sum_{i=1}^n \\frac{n^{-1\/2} i \\exp(-i\n d\/\\sqrt{n})}{1 + \\exp(-i d \/ \\sqrt{n})} \\frac{1}{\\sqrt{n}} \\\\\n & \\to & \\int_0^{\\infty} \\frac{y e^{-d y}}{1 + e^{- d y}} dy \\\\\n & = & \\frac{1}{d^2} \\int_0^1 \\frac{- \\log v}{1+v} dv \\\\\n & = & \\frac{\\pi^2}{12 d^2}.\n \\end{eqnarray*}\n Hence to satisfy ${\\Bbb E} T_n \\sim n$, we pick $d = \\pi\/\\sqrt{12}$, so that\n \\begin{equation}\\label{choosex-select}\n x = \\exp(-\\pi\/ \\sqrt{12 n}).\n \\end{equation}\n From (\\ref{multi-vartn}), it follows by a similar calculation that\n \\begin{eqnarray*}\n n^{-3\/2}\\, \\sigma_n^2 & \\to & \\int_0^{\\infty} \\frac{y^2 e^{-d y}}\n {(1 + e^{- d y})^2} dy \\\\\n & = & \\frac{1}{d^3} \\int_0^1 \\left(\\frac{- \\log v}{1+v}\\right)^2 dv \\\\\n & = & \\frac{2}{d}.\n \\end{eqnarray*}\n For the choice of $x$ in (\\ref{choosex-select}), we see that\n \\begin{equation}\\label{vartn-selectpart}\n \\sigma_n^2 \\sim \\frac{4 \\sqrt{3}}{\\pi} n^{3\/2}.\n \\end{equation}\n  \n To see how easy the heuristic for choosing $x$ can be, consider\n partitions of the integer $n$ with all parts distinct and odd.  Compared\n to the above calculations, we are simply leaving out every other term,\n so that $n^{-1} \\B ET_n \\rightarrow \\pi^2\/(24d^2)$, and we prescribe\n using $x= \\exp(-\\pi\/ \\sqrt{24n})$.  As with unrestricted partitions, using\n the appropriate $x$ for either partitions with distinct parts or\n partitions with distinct odd parts, we believe that the unconditioned\n process $\\b Z_B$ is a good approximation for the combinatorial process\n $\\b C_B$, in the total variation sense, if and only if $b\/\\sqrt{n}$ is\n small and $c\/\\sqrt{n}$ is large, for \n  $B=\\{1,2,\\ldots,b \\} \\cup \\{c,c+1,\\ldots,n \\}$.\n For $B$ of the complementary form\n $B=\\{b,b+1,\\ldots,c \\}$ with $b<c$ both of the order of $\\sqrt{n}$, \n  the heuristic suggests that $d_{TV}(\\b C_B, \\b Z_B)\n $ is small if, and only if, $(c-b)$ is small relative to $\\sqrt{n}$.\n \n \\subsection{A quantitative heuristic}\\label{sect4.2}\n\n In several examples, the ${\\Bbb Z}_+$-valued random variables $T_n$,\n appropriately centered and rescaled, converge in distribution to a continuous \n limit $X$ having a density $f$ on ${\\Bbb R}$. For illustration, we \ndescribe the important class of cases in which \n \\begin{equation}\\label{rescale0}\n \\frac{T_n}{n} \\Rightarrow X.\n \\end{equation}\nA local limit heuristic suggests the approximation\n\\begin{equation}\\label{local1}\n{\\Bbb P}(T_n = n)  \\doteq \\frac{f(1)}{n},\n\\end{equation}\nwhere the sense of the approximation $\\doteq$ is deliberately vague.\nAssuming that $B$ is small, so that $R\/n \\mbox{$\\rightarrow_P\\ $} 0$, we also have $S\/n\n\\Rightarrow X$.\nFor $0 \\leq k \\leq n$, the local limit heuristic gives\n$$\n{\\Bbb P}(S = n-k)  \\doteq \\frac{1}{n}\\,f\\left(1 - \\frac{k}{n}\\right),\n$$\nand a Taylor expansion further simplifies this to \n\\begin{equation}\\label{local2}\n{\\Bbb P}(S = n-k) \\doteq \\frac{1}{n} \\left(f(1) - \\frac{k}{n} f^\\prime(1-\n)\\right).\n\\end{equation}\n\nUsing these approximations in the total variation  formula (\\ref{tvcalc}) gives\n \\begin{eqnarray*}\n d_{TV}(\\b C_B,\\b Z_B) & = & \\frac{1}{2} \\sum_{k = 0}^n {\\Bbb P}(R=k) \\left|\n 1 - \\frac{{\\Bbb P}(S=n-k)}{{\\Bbb P}(T_n = n)}\\right| + \\frac{1}{2}{\\Bbb P}(R > n)\\\\\n &&  \\\\\n & \\doteq & \\frac{1}{2} \\sum_{k \\geq 0} {\\Bbb P}(R=k) \\left| 1 - \\frac{\n n^{-1}  (f(1) - n^{-1} k f^{\\prime}(1-)}\n {n^{-1} f(1)}\\right| \\\\\n &&  \\\\\n & = & \\frac{1}{2} \\frac{|f^{\\prime}(1-)|}{f(1)} \n \\frac{{\\Bbb E} |R|}{n}. \n \\end{eqnarray*}\n However, this approximation ignores the essential feature that\n$d_{TV}(\\mu,\\nu) = \\frac{1}{2} |\\mu - \\nu|$, where the signed measure\n $\\mu -\\nu$ has net mass zero.\nThus, even though $f(1)\/n$ is the natural approximation for ${\\Bbb P}(T_n =\nn)$, it is important to use a more complicated heuristic in which the \napproximation for $T$ is the\nconvolution of the distribution of $R$ and our approximation for the\ndistribution of $S$. Thus\n\\begin{eqnarray}\n{\\Bbb P}(T = n) & = & \\sum_{k =0}^n {\\Bbb P}(R=k) {\\Bbb P}(S = n-k) \\nonumber\\\\\n&&\\nonumber\\\\\n& \\doteq & \\sum_{k \\geq 0} {\\Bbb P}(R = k) \\frac{1}{n} \\left(f(1) - \\frac{k}{n}\nf^\\prime(1-) \\right) \\nonumber \\\\\n&&\\nonumber \\\\\n& = & \\frac{1}{n} \\left( f(1) - \\frac{{\\Bbb E} R}{n} f^\\prime(1-)\\right).\n\\label{pt=napprox}\n\\end{eqnarray}\nUsing this approximation,\n \\begin{eqnarray}\n d_{TV}(\\b C_B,\\b Z_B) & = & \\frac{1}{2} \\sum_{k = 0}^n {\\Bbb P}(R=k) \\left|\n 1 - \\frac{{\\Bbb P}(S=n-k)}{{\\Bbb P}(T_n = n)}\\right| + \\frac{1}{2} {\\Bbb P}(R > n)\\nonumber\\\\\n && \\nonumber \\\\\n & \\doteq & \\frac{1}{2} \\sum_{k \\geq 0} {\\Bbb P}(R=k) \\left| 1 - \\frac{\n n^{-1}  (f(1) - n^{-1} k f^{\\prime}(1-))}\n {n^{-1} (f(1) - n^{-1} {\\Bbb E} R f^\\prime(1-))}\\right| \\nonumber \\\\\n && \\nonumber \\\\\n& = & \\frac{1}{2} \\sum_{k \\geq 0} {\\Bbb P}(R=k) \\left| \\frac{n^{-1} (k - {\\Bbb E} R)\nf^\\prime(1-)}{f(1) - n^{-1} {\\Bbb E} R f^\\prime(1-)}\\right| \\nonumber \\\\\n&& \\nonumber \\\\\n& = & \\frac{1}{2n} |f^\\prime(1-)| \\  {\\Bbb E} | R - {\\Bbb E} R|\n\\ |f(1) - n^{-1} {\\Bbb E} R f^\\prime(1-)|^{-1} \\nonumber \\\\\n&& \\nonumber \\\\\n& \\doteq & \\frac{1}{2n} \\frac{|f^\\prime(1-)|}{f(1)} {\\Bbb E} | R - {\\Bbb E} R|.\n  \\label{approx2}\n \\end{eqnarray}\n\nAs a plausibility check, we note that the alternative approximation\nusing ${\\Bbb P}(T_n = n) \\doteq \\frac{1}{n} f(1)$ and $S \\doteq T - {\\Bbb E} R$,\nso that ${\\Bbb P}(S = n-k) \\doteq {\\Bbb P}(T = n + {\\Bbb E} R - k) \\doteq \\frac{1}{n}\nf(1 - \\frac{k-{\\Bbb E} R}{n})$, also satisfies the convolutional property, and\nleads to the same first order result as (\\ref{approx2}).\n\n One possible specific interpretation  of the approximation in (\\ref{approx2})\n would be the following pair of statements, giving a decay rate for\n$d_{TV}$, for fixed $B$, as $n \\to\n\\infty$.\n \n {\\it If $T_n\/n \\Rightarrow X$, and $X$ has density $f$ with\n$f^\\prime(1-) \\neq 0$, then}\n \\begin{equation}\\label{approx3}\n d_{TV}(\\b C_B,\\b Z_B) \\sim \\frac{1}{2} \\frac{|f^{\\prime}(1-)|}{f(1)}\n \\frac{{\\Bbb E}|R - {\\Bbb E} R|}{n},\n \\end{equation}\n \n {\\it If $T_n\/n \\Rightarrow X$, and $X$ has density $f$ with\n$f^\\prime(1-) = 0$, then}\n \\begin{equation}\\label{approx4}\n d_{TV}(\\b C_B,\\b Z_B) = o\\left(\\frac{1}{n}\\right).\n \\end{equation}\n \nFor the more general case in which there are constants $s_n$ such that\n$$\n\\frac{T_n - n}{s_n} \\Rightarrow X\n$$\nwhere $X$ has density $f$, these statements are to be replaced by\n\\begin{equation}\\label{approx3gen}\n d_{TV}(\\b C_B,\\b Z_B) \\sim \\frac{1}{2} \\frac{|f^{\\prime}(0-)|}{f(0)}\n \\frac{{\\Bbb E}|R - {\\Bbb E} R|}{s_n},\\ {\\rm\\ if\\ }f^\\prime(0-) \\neq 0,\n \\end{equation}\nand\n \\begin{equation}\\label{approx4gen}\n d_{TV}(\\b C_B,\\b Z_B) = o\\left(\\frac{1}{s_n}\\right),\\ {\\rm\\ if\\ }\nf^\\prime(0-) = 0.\n \\end{equation}\nFor partitions of an integer and for partitions of a set, a good choice\nfor $s_n$ is the standard deviation $\\sigma_n$\n with asymptotics given by (\\ref{vartn-multipart}) and\n(\\ref{Moser2}), and $X$ is normally distributed, so that\n(\\ref{approx4gen}) should apply.\n\n Observe that for two fixed sets $B, B^{\\prime}$ the approximation  in\n (\\ref{approx3}) or (\\ref{approx3gen}) has as a corollary the\nstatement  that if $f^\\prime(0-) \\neq 0$ then\nas $n \\to \\infty$,\n $$\n \\frac{d_{TV}(\\b C_B,\\b Z_B)}{d_{TV}(\\b C_{B^{\\prime}},\\b Z_{B^{\\prime}})} \\to\n \\frac{{\\Bbb E} |R_B - {\\Bbb E} R_B|}{ {\\Bbb E} |R_{B^{\\prime}} - {\\Bbb E} R_{B^{\\prime}}|}.\n $$\n \n By the Cauchy-Schwarz inequality,\n ${\\Bbb E} |R_B - {\\Bbb E} R_B| \\leq \\sigma_B$, so another rigorous version\n of the heuristic in (\\ref{approx2}) would be the statement that as $n \\to\n \\infty$, $d_{TV}(\\b C_B, \\b Z_B) = O( \\sigma_B\/ \\sigma_n)$ uniformly in\n $B$; that is\n \\begin{equation}\\label{approx5}\n \\lim_{n \\to \\infty} \\sup_{B \\subset [n]} \\left( d_{TV}(\\b C_B, \\b Z_B)\n \\frac{\\sigma_n}{\\sigma_B}\\right) < \\infty.\n \\end{equation}\n Note that (\\ref{approx5}) is not embarrassed by the largest possible\n $B$, namely $B = [n]$, since $d_{TV}(\\cdot,\\cdot) \\leq 1$.\n \n\n\n\n \n\\subsection{Examples with a limit process: the logarithmic class}\n\\label{logsect}\n \nThe previous section suggests that the limit law of $T_n \/ n$ plays a key\nrole in analyzing the accuracy  of the approximation of certain\ncombinatorial structures by independent processes. \nThe logarithmic class consists of those assemblies which satisfy\n(\\ref{assembly-hyp}), and those multisets and selections which satisfy\n(\\ref{multiset-hyp}).  All of these, with the appropriate constant\nchoice of $x$, satisfy \n\\begin{equation}\\label{Zcond}\ni \\B EZ_i \\rightarrow \\kappa,\\ i {\\Bbb P}(Z_i=1)\n\\rightarrow \\kappa {\\rm \\ for\\ some\\ } \\kappa >0.\n\\end{equation}\nLemma \\ref{tnn-mul} below shows that, for $Z_i$ satisfying\n(\\ref{Zcond}), and $T_n=Z_1+2Z_2+\\cdots+nZ_n$, the limit distribution of\n$T_n \/n$ depends only on the parameter $\\kappa$.\n\nLet $d_W$ be the $L_1$ Wasserstein distance\nbetween distributions, which  can be defined, in the same\nspirit as (\\ref{tvcouple}), by \n$$\nd_W(X,Y) =\n\\min_{couplings} {\\Bbb E} |X-Y|.\n$$\nFor ${\\Bbb Z}^+$-valued random variables,\n$d_W$ is easily computed via \n$$\nd_W(X,Y)=\\sum_{i \\geq 1} |{\\Bbb P}(X\\geq i)-{\\Bbb P}(Y\\geq i)|,\n$$\nand when $X$ is stochastically larger than $Y$, so that the absolute\nvalues above do nothing, this further simplifies to $d_W(X,Y)={\\Bbb E} X-{\\Bbb E}\nY$. Note that for integer-valued random values, $d_W \\geq d_{TV}$.  \n\nLet  $\\tilde Z_i$ be\n Bernoulli with parameter $\\kappa\/i \\wedge 1$, and let \n$Z_i^*$ be Poisson with mean $\\kappa\/i$. It is easy to check that the\n condition (\\ref{Zcond}) is equivalent to \n$d_W(Z_i,\\tilde Z_i)=o(1\/i)$. Since $d_W(\\tilde Z_i,Z_i^*)=o(1\/i)$, the triangle\ninequality implies that\nthe condition (\\ref{Zcond}) is also equivalent to \n$d_W(Z_i,Z_i^*)=o(1\/i)$. \n\nFor the class of assemblies that satisfy the condition\n(\\ref{assembly-hyp}), we use $x=y^{-1}$ and ${\\Bbb E} Z_i=m_i x^i\/i!$, so\nthat ${\\Bbb E} Z_i \\sim \\kappa\/i$.  Lemma \\ref{tnn-ass} applies directly;\nfor Poisson random variables (\\ref{Zcond}) is equivalent to $\\B E Z_i\n\\sim \\kappa\/i$, so Lemma \\ref{tnn-mul} also applies. \nFor multisets  and selections \nsatisfying the hypothesis (\\ref{multiset-hyp}), it is easy to show\nthat (\\ref{Zcond}) holds.\n\n\\begin{lemma}\\label{tnn-ass}\nIf $Z_j$ are\nindependent Poisson random variables with ${\\Bbb E} Z_j = \\lambda_j \\sim\n\\kappa \/j$ for some constant $\\kappa >0$, and\n$T_n = \\sum_{j=1}^n j Z_j$, then \n\\begin{equation}\\label{limitlaw}\nn^{-1} T_n \\Rightarrow X_\\kappa,\\ n \\to \\infty\n\\end{equation}\nand $X_\\kappa$ has Laplace transform \n\\begin{equation}\\label{limitlt}\n\\psi(s) \\equiv {\\Bbb E} e^{- s X_\\kappa} = \\exp\\left( - \\kappa \\int_0^1 \n(1 - e^{-s x}) \\frac{dx}{x} \\right).\n\\end{equation}\n\\end{lemma}\n\n\\noindent{\\bf Proof\\ \\ } By direct calculation,\n\\begin{eqnarray*}\n\\log {\\Bbb E} e^{- s T_n \/ n} & = & - \\sum_{j = 1}^n \\lambda_j (1 -\ne^{-j s \/ n}) \\\\\n&&\\\\\n& = & - \\sum_{j = 1}^n \\frac{\\kappa}{j} (1 - e^{-j s \/ n}) +\n\\sum_{j = 1}^n \\left( \\frac{\\kappa}{j} - \\lambda_j \\right) (1 -\ne^{-j s \/ n})\n\\end{eqnarray*}\nClearly, the first term on the right converges to $-\\kappa \\int_0^1(1-\ne^{-sx})\\frac{dx}{x}$. That\nthe second term is $o(1)$ follows by observing that $\\lambda_j -\n\\kappa \/ j = o(j^{-1})$, and comparing to the first sum.\n\\hfill \\mbox{\\rule{0.5em}{0.5em}}\n\n\n\\begin{lemma}\\label{tnn-mul}\nFor $i=1,2,\\ldots$, let $Z_i$ be positive integer-valued random\nvariables satisfying the conditions in (\\ref{Zcond}).\nIf $T_n = \\sum_{j=1}^n j Z_j$, then \n\\begin{equation}\\label{limitlawmul}\nn^{-1} T_n \\Rightarrow X_\\kappa,\\ n \\to \\infty\n\\end{equation}\nand $X_\\kappa$ has Laplace transform given in (\\ref{limitlt}).\n\\end{lemma}\n\n\\noindent{\\bf Proof\\ \\ } Construct independent Bernoulli random variables $\\tilde Z_i =\nZ_i \\wedge 1$. Clearly $\\tilde Z_i \\leq Z_i$ and\n${\\Bbb P}(Z_i = 1) \\leq {\\Bbb E} \\tilde Z_i \\leq {\\Bbb E} Z_i$. It follows that \n$i {\\Bbb E} \\tilde Z_i \\to \\kappa$. Therefore\n$$\ni|{\\Bbb E} Z_i - {\\Bbb E} \\tilde Z_i| = i({\\Bbb E} Z_i - {\\Bbb E} \\tilde Z_i) \\to 0.\n$$\nHence if $\\tilde T_n = \\tilde Z_1 + \\cdots + n \\tilde Z_n$,\n\\begin{equation}\\label{useme}\n{\\Bbb E} \\left|\\frac{T_n}{n} - \\frac{\\tilde T_n}{n} \\right| \\to 0.\n\\end{equation}\nIt remains to show that $n^{-1} \\tilde T_n \\Rightarrow X_{\\kappa}$.\n\nFor $i=1,2,\\ldots$, let $Z_i^*$ be independent Poisson random variables\nsatisfying $p_i \\equiv {\\Bbb E} Z_i^* = {\\Bbb E} \\tilde Z_i \\sim \\kappa\/i.$ \nWe may construct\n$Z_i^*$ in such a way that for each $i$\n$$\n{\\Bbb E} |\\tilde Z_i - Z_i^*| = d_W(\\tilde Z_i, Z_i^*),\n$$\nwhere $d_W$ denotes Wasserstein $L_1$ distance. But if $X$ is Bernoulli\nwith parameter $p$ and $Y$ is Poisson with parameter $p$, then a simple\ncalculation shows that $d_W(X,Y) = 2(p - 1 + e^{-p}) \\leq p^2$.\nHence \n$$\nn^{-1} {\\Bbb E}|\\tilde T_n - T_n^*| \\leq n^{-1} \\sum_{i=1}^n i p_i^2 \\to 0.\n$$\nIt follows that $n^{-1} \\tilde T_n$ has the same limit law as $n^{-1}\nT_n^*$, which is that of $X_{\\kappa}$ by Lemma\n\\ref{tnn-ass}. \\hfill\\mbox{\\rule{0.5em}{0.5em}}\n\nThe random variable $X_\\kappa$ has appeared in several guises before, not\nleast as part of the description of the density of points in a \nPoisson--Dirichlet process. See Watterson (1976), Vershik and Shmidt\n(1977), Ignatov (1982), Griffiths (1988) and Ethier and Kurtz (1986) and \nthe references contained therein. \nFor our purposes, it is enough to record that the density $g(\\cdot)$ of\n$X_\\kappa$ is known explicitly on the interval $[0,1]$:\n\\begin{equation}\\label{ignatov}\ng(z) = \\frac{e^{- \\gamma \\kappa}}{\\Gamma(\\kappa)} z^{\\kappa - 1},\\ 0\n\\leq z \\leq 1,\n\\end{equation}\nwhere $\\gamma$ is Euler's constant.\nFrom (\\ref{ignatov}) follows the fact that\n\\begin{equation}\\label{ignatov1}\n\\frac{g^\\prime(1-)}{g(1)} = \\kappa - 1.\n\\end{equation}\nWe may now combine the previous results with (\\ref{approx2}) and\n(\\ref{mean-varass}) to rephrase the asymptotic\nbehavior of $d_{TV}(\\mbox{\\boldmath $C$}_B,\\mbox{\\boldmath $Z$}_B)$ in (\\ref{approx3}) and\n(\\ref{approx4}) as follows. For any assembly satisfying\n(\\ref{assembly-hyp}), or for any multiset or selection satisfying\n(\\ref{multiset-hyp}), we should have the following decay rates, \nfor any fixed $B$, as $n \\rightarrow \\infty$.\n\n {\\it In the case $\\kappa \\neq 1$}\n \\begin{equation}\\label{approx3b}\n d_{TV}(\\b C_B,\\b Z_B) \\sim \\frac{1}{2} |\\kappa - 1|\n \\frac{{\\Bbb E}|R - {\\Bbb E} R|}{n},\n \\end{equation}\n \n {\\it In the case $\\kappa = 1$}\n \\begin{equation}\\label{approx4b}\n d_{TV}(\\b C_B,\\b Z_B) = o\\left(\\frac{1}{n}\\right).\n \\end{equation}\n\n\n For a class of examples known as the Ewens\nsampling formula, described in Section \\ref{esfsect},\nand for $B$ of the form \n$B = \\{1,2,\\ldots,b\\}$, (\\ref{approx3b}) is\nproved in Arratia, Stark and Tavar\\'e (1994). The analogous result for \nrandom mappings, in\nwhich   $\\kappa = 1\/2$, and other assemblies that can be approximated by the\nEwens sampling formula, may also be found there.\nFor the corresponding results for multisets and selections, see Stark\n(1994b).\n\nThe statement (\\ref{approx4b}) has been established for random\npermutations by Arratia and Tavar\\'e (1992), where it is shown inter\nalia that for $B = \\{1,2,\\ldots,b\\}$, $d_{TV}(\\mbox{\\boldmath $C$}_B,\\mbox{\\boldmath $Z$}_B) \\leq F(n\/b)$,\nwhere $\\log F(x) \\sim -x \\log x$ as $x \\to \\infty$. For the case of\nrandom polynomials over a finite field, Arratia, Barbour and Tavar\\'e\n(1993) established that $d_{TV}(\\mbox{\\boldmath $C$}_B,\\mbox{\\boldmath $Z$}_B) = O(b \\exp(-c n \/ b))$,\nwhere $c = \\frac{1}{2}\\log(4\/3)$.\n\nAmong the class of assemblies in the logarithmic class, weak convergence \n(in ${\\Bbb R}^\\infty$) of the component counting process to\nthe appropriate Poisson process has been established for random\npermutations by Goncharov (1944), for random mappings by Kolchin (1976),\nand for the Ewens sampling formula by Arratia, Barbour and Tavar\\'e\n(1992). For multisets in the logarithmic class, this has been\nestablished for random polynomials by Diaconis, McGrath and Pitman\n(1994) and Arratia, Barbour and Tavar\\'e (1993), and for random mapping\npatterns by Mutafciev (1988).\n\n\n\n  \\section{Non-uniqueness in the choice of the parameter $x$}\n \\label{sect5}\n\n An appropriate choice of $x = x(n)$ for good approximation is not\n unique. \nAn obvious candidate is that $x$ which maximizes ${\\Bbb P}(T_n = n)$, which\nis also that $x$ for which ${\\Bbb E} T_n = n$. This can be seen by\ndifferentiating $\\log {\\Bbb P}(T_n = n)$ in formulas (24) - (26) and\ncomparing to ${\\Bbb E} T_n$ from formulas (11) - (13); at the general level\nthis is the observation that ${\\Bbb P}(T=t)$ in (19) is maximized by that $x$\nfor which ${\\Bbb E} T = t$. Nevertheless, the obvious candidate is not always\nthe best one.\nWe discuss here two qualitatively different examples: the\nlogarithmic class, and partitions of a set.\n\n\\subsection{The Ewens sampling formula}\\label{esfsect}\n\nThe central object in the logarithmic class is the Ewens sampling\nformula (ESF). This is the family of distributions with parameter\n$\\kappa > 0$ given by\n(\\ref{equaldist}), where the $Z_i$ are independent Poisson random\nvariables with ${\\Bbb E} Z_i = \\kappa\/i$, or more generally, with\n\\begin{equation}\\label{esfmeans}\n\\lambda_i \\equiv {\\Bbb E} Z_i = \\frac{\\kappa x^i}{i},\n\\end{equation}\nthe conditional distribution being unaffected by the choice of $x >\n0$.\nFor $\\kappa = 1$, the ESF is the distribution of cycle counts for a\nuniformly chosen random permutation. For $\\kappa \\ne 1$, the ESF \ncan be viewed as the nonuniform measure\non permutations with sampling bias proportional to $\\kappa^{{\\rm \\#\\\ncycles}}$; see Section \\ref{sect8} for details. \nThe ESF arose first in the context\nof population genetics (Ewens, 1972), and is given explicitly by\n\\begin{equation}\\label{esfdef}\n{\\Bbb P}(C_1(n)=a_1,\\ldots,C_n(n) = a_n) =\n\\mbox{\\bf 1}(\\sum_{l=1}^n l a_l = n) \\ \\frac{n!}{\\kappa_{(n)}} \\ \\prod_{i=1}^n\n\\left( \\frac{\\kappa}{i} \\right)^{a_i} \\frac{1}{a_i!} .\n\\end{equation}\nThe ESF corresponds to (\\ref{assembly-hyp}) with $y = 1$ and\nthe asymptotic relation in $i$ replaced by equality. It is useful in\ndescribing all assemblies, multisets and selections in the logarithmic\nclass; see Arratia, Barbour and Tavar\\'e (1994) for further details.\n\nFor irrational $\\kappa$ the ESF cannot  be realized as\na uniform measure on a class of combinatorial objects.  For rational\n$\\kappa =r\/s$ with integers $r>0, s>0$, there are at least two\npossibilities.  First, comparing  \n(\\ref{equaldist}) with $\\B EZ_i = \\kappa\/i$, and (\\ref{Zassembly})\nwith $\\B EZ_i= m_i x^i \/i!,$ for any choice $x>0$, we take $x=1\/s$ to \nsee that the ESF\nis the uniform measure on the assembly with \n$m_i =r(i-1)!s^{i-1}$.  One interpretation of this is permutations on\nintegers, enriched by coloring each cycle with one of $r$ possible\ncolors, and coloring each element of each cycle, except the smallest,\nwith one of $s$ colors.  For a second construction, we use a device from\nStark (1994a).  Consider permutations of $ns$ objects, \nin which all\ncycle lengths must be multiples of $s$.  Formally, this is the assembly\non $ns$ objects, with $m_i=(i-1)!\\mbox{\\bf 1} (s|i)$, so that\n$(C_1,C_2,\\ldots,C_{ns}) \\stackrel{\\mbox{\\small d}}{=}\n(Z_1,Z_2,\\ldots,Z_{ns}|Z_1+2Z_2+\\cdots+nsZ_{ns}=ns)$, where $Z_i$ is\nPoisson with $\\B EZ_i= \\mbox{\\bf 1}(s|i) \\ \/i$.  Since those $C_i$ and\n$Z_i$ for which $s$ does not divide $i$ are identically zero, we\nconsider $C_i^*\\equiv C_{is}, \\ Z_i^* \\equiv Z_{is}$, and $T_n^* \\equiv\nZ_1^*+2Z_2^*+\\cdots+nZ_n^* = \\frac{1}{s}(Z_1+2Z_2+\\cdots+nsZ_{ns})$. \nWe have $(C_1^*,\\ldots,C_n^*) \\stackrel{\\mbox{\\small d}}{=} (Z_1^*,\\ldots,Z_n^* | T_n^* = n)$, and the\n$Z_i^*$ are independent Poisson with $\\B E Z_i^*=1\/(si)$.  Thus\nthe distribution of $(C_1^*(n),\\ldots,C_n^*(n))$ is the ESF with\n$\\kappa=1\/s$.  To change this to $\\kappa=r\/s$, we need only\ncolor each cycle with one of $r$ possible colors, so that $m_i = r (i-\n1)! \\mbox{\\bf 1}(s|i), \\B EZ_i = r \\mbox{\\bf 1} (s\/i)\\ \/i$, and $\\B EZ_i^* =\nr\/(si)$.  To summarize our second construction of the ESF with\n$\\kappa =r\/s$, let $C_i^*(n)$ be the number of cycles of length $si$ in\na random permutation of $ns$ objects, requiring that all cycle lengths\nbe multiples of $s$, and assigning one of $r$ possible colors to each\ncycle. \n\nFor comparing the ESF to the unconditioned, independent process\n$(Z_1,\\ldots,$ $Z_n)$ it is interesting to consider the role of varying $x$.\n The choice $x = 1$ in (\\ref{esfmeans}), so that ${\\Bbb E} Z_i = \\kappa \/i$,\nyields ${\\Bbb E} T_n = \\kappa n$, and $\\sigma_n \\sim n \\sqrt{\\kappa\/2}$.\n In the case $\\kappa \\ne 1$ the discrepancy between ${\\Bbb E} T_n$ and the\n goal $n$ is a bounded multiple of $\\sigma_n$. This is close enough\n for good approximation, in the sense that\n $(C_1(n),\\ldots,C_n(n),0,\\ldots)$ $\\Rightarrow (Z_1,Z_2,\\ldots)$.\n This, together with a $O(b\/n)$ bound on\n $d_{TV}(C_1(n),\\ldots,C_b(n)),(Z_1,\\ldots,Z_b))$ that is  uniform in $1 \\leq b\n \\leq n$, is proved in {\\rm Arratia, Barbour and Tavar\\'e (1992)} \\ by exploiting a coupling\n based on Feller (1945). This coupling\n provided even stronger information whose utility is discussed in {\\rm Arratia and Tavar\\'e (1992b)}.\n Barbour (1992) showed that the $O(b\/n)$ bound above cannot be replaced\n by $o(b\/n)$ for $x=1$, $\\kappa \\ne 1$. \n\n\n For the case of independent $Z_i^{\\prime}$ which are Poisson with means\nvarying with $n$ given by\n $$\n {\\Bbb E} Z_i^{\\prime} = {\\Bbb E} C_i(n) = \\frac{\\kappa}{i} \\frac{n (n-\n 1)\\cdots(n-i+1)}{(\\kappa+n-i)\\cdots(\\kappa+n-1)},\n $$\n Barbour (1992) showed that\n $d_{TV}(C_1(n),\\ldots,C_b(n)),(Z_1^{\\prime},\n \\ldots,Z_b^{\\prime})) =\n O((b\/n)^2)$, uniformly in $1 \\leq b \\leq n$. Observe that with this\n choice of Poisson parameters, ${\\Bbb E} T^\\prime_n \\sim \\kappa n$ but it is \n{\\it not} \nthe case \nthat $(C_1(n),\\ldots,C_n(n)) \\stackrel{\\mbox{\\small d}}{=} (Z_1^\\prime,\\ldots,Z_n^\\prime | \nT^\\prime_n = n)$.\n\nIf we are willing to use coordinates $Z_i \\equiv Z_i(n)$ whose means\nvary with $n$, we can still have the conditional relation\n(\\ref{equaldist}) by using $x = x(n)$ in (\\ref{esfmeans}). An\nappealing family of choices is given by $x = \\exp(- c\/n)$, since this\nyields for $c \\neq 0$\n\\begin{equation}\\label{newmean}\n{\\Bbb E} T_n = \\sum_{i=1}^n i \\lambda_i = \\sum_{i=1}^n i \\frac{\\kappa}{i}\ne^{-i c \/ n} \\sim n \\frac{\\kappa(1 - e^{-c})}{c}.\n\\end{equation}\n By choosing $c \\equiv\n c(\\kappa)$ as the solution of $\\kappa = c\/(1-e^{-c})$, we can make ${\\Bbb E}\n T_n \\sim n$, and this should provide a closer approximation than the\nchoice $c=0, x=1$. \n However an even better choice of $c$ is available. We explore this in\nthe next section.\n \n \\subsection{More accurate approximations to the logarithmic class}\n \\label{sect5.2}\n\nFor assemblies, multisets, and selections in the logarithmic class\ndiscussed in Section \\ref{logsect}, as\nwell as for the ESF, the choice  of $x$ proportional to $\\exp(-c\/n)$ is\ninteresting. In this situation, the limit law of $T_n\/n$ depends only on\nthe parameters $\\kappa$ and $c$. Properties of this limit law lead to an\noptimal choice for $c$.\n\nThe following lemma applies to assemblies that satisfy the condition\n(\\ref{assembly-hyp}), and to the ESF by taking $m_i = \\kappa (i-1)!,\ny=1$, the $m_i$ not necessarily being integers.\n\n\\begin{lemma}\\label{tnn-assnew}\nAssume that $m_i \\geq 0$ satisfies \n$m_i\/i! \\sim \\kappa y^i \/ i$ for constants $y \\geq 1, \\kappa > 0$, \nand set $x = e^{- c \/ n} y^{-1}$ for constant $c \\in {\\Bbb R}$. If $Z_j\n\\equiv Z_j(n)$ are\nindependent Poisson random variables with ${\\Bbb E} Z_j = m_j x^j \/ j!$, and\n$T_n = \\sum_{j=1}^n j Z_j$, then \n\\begin{equation}\\label{limitlaw1}\nn^{-1} T_n \\Rightarrow X_{\\kappa,c},\\ n \\to \\infty\n\\end{equation}\nand $X_{\\kappa,c}$ has Laplace transform \n\\begin{equation}\\label{limitlt1}\n\\psi_c(s) \\equiv {\\Bbb E} e^{- s X_{\\kappa,c}} = \\exp\\left( - \\kappa \\int_0^1 \n(1 - e^{-s x}) \\frac{e^{-c x}}{x} dx \\right).\n\\end{equation}\n\\end{lemma}\n\n\n\\noindent{\\bf Proof\\ \\ } As in Lemma \\ref{tnn-ass}, calculate the limit of the \nlog Laplace transform.\\hfill \\mbox{\\rule{0.5em}{0.5em}}\n\nNext we prove that the same limit law holds \nfor multisets or selections satisfying the hypothesis (\\ref{multiset-hyp}). \n\\begin{lemma}\\label{tnn-mulnew}\nAssume that the multiset (or selection) satisfies (\\ref{multiset-hyp}):\n$m_i \\sim \\kappa y^i \/ i$ for constants $y \\geq 1, \\kappa >0$,\n and set $x = e^{ - c \/ n} y^{-1}$. If $Z_j\n\\equiv Z_j(n)$ are\nindependent negative binomial random variables with parameters $m_j$ and\n$ x^j$ (respectively, binomial with parameters $m_j$ and $x^j\/(1+x^j))$ \nand $T_n = \\sum_{j=1}^n j Z_j$, then \n\\begin{equation}\\label{limitlawmul2}\nn^{-1} T_n \\Rightarrow X_{\\kappa,c},\\ n \\to \\infty\n\\end{equation}\nand $X_{\\kappa,c}$ has Laplace transform given in (\\ref{limitlt1}).\n\\end{lemma}\n\n\\noindent{\\bf Remark: } For the case of multisets, we assume that $x < 1$.\n\n\\noindent{\\bf Proof\\ \\ } Observe first that in either case, if $b = o(n)$, then \n$n^{-1} {\\Bbb E} T_{0b} \\rightarrow 0$, so that $n^{-1} T_{0b} \\mbox{$\\rightarrow_P\\ $} 0$ as\n$n \\to \\infty$. Let $\\tilde Z_j$ be independent Poisson random variables\nwith ${\\Bbb E} \\tilde Z_j = m_j x^j$, and write \n$\\tilde T_n = \\sum_{j=1}^n j\n\\tilde Z_j$, $\\tilde T_{bn} = \\sum_{j=b+1}^n j \\tilde Z_j$. We show\nthat for $b = o(n)$, $T_{bn}\/n$ and $\\tilde T_{bn} \/ n$ have the same limit \nlaw, which completes the proof since by Lemma \\ref{tnn-assnew},\n$\\tilde T_{bn} \/ n \\Rightarrow X_{\\kappa,c}$.\nWe will use the notation NB, Po, and Geom to denote the negative\nbinomial, Poisson and geometric distributions with the indicated\nparameters.\n\nFor the multiset case, notice that\n\\begin{eqnarray*}\nd_{TV}(T_{bn},\\tilde T_{bn}) & \\leq & d_{TV}((Z_{b+1},\\ldots,Z_n),\n(\\tilde Z_{b+1},\\ldots,\\tilde Z_n)) \\\\\n& \\leq & \\sum_{b+1}^n d_{TV}(Z_j,\\tilde Z_j).\n\\end{eqnarray*}\nTo estimate each summand, we have\n\\begin{eqnarray}\nd_{TV}(Z_j, \\tilde Z_j) & = & d_{TV}({\\rm NB}(m_j, x^j), {\\rm Po}(m_j\nx^j)) \\nonumber \\\\\n& \\leq & m_j d_{TV}( {\\rm Geom}(x^j), {\\rm Po}(x^j)) \\nonumber\\\\\n& \\leq & 2 m_j x^{2j}. \\label{qandd1}\n\\end{eqnarray}\nThe bound in (\\ref{qandd1}) follows from the fact that\n$d_{TV}({\\rm Geom}(p), {\\rm Be}(p)) = p^2$ and  $d_{TV}({\\rm Be}(p), {\\rm\nPo}(p)) = p(1-e^{-p}) \\leq p^2$, so that\n$d_{TV}({\\rm Geom}(p),{\\rm Po}(p)) \\leq \nd_{TV}({\\rm Geom}(p), {\\rm Be}(p))$ $+\\ d_{TV}({\\rm Be}(p), {\\rm Po}(p)) \n\\leq 2p^2$, a result we apply with $p = x^j$. Hence\n$$\nd_{TV}(T_{bn}, \\tilde T_{bn}) \\leq 2 \\sum_{j=b+1}^n (m_j x^j) x^j =\nO(y^{-b}\/b).\n$$\nChoosing $b \\to \\infty, b = o(n)$ completes the proof for multisets.\n\nFor the selection case, (\\ref{qandd1}) may be replaced by\n$$\nd_{TV}(Z_j, \\tilde Z_j) \\leq m_j d_{TV}({\\rm Be}(x^j\/(1 + x^j)), {\\rm\nPo}(x^j)) \\leq 2 m_j x^{2j}.\n$$\nThe last estimate following from the observation that $d_{TV}({\\rm\nBe}(p\/(1 + p)), {\\rm Be}(p)) = p^2\/(1+p)$, so that\n$d_{TV}({\\rm Be}(p\/(1 + p)),{\\rm Po}(p)) \\leq d_{TV}({\\rm\nBe}(p\/(1 + p)), {\\rm Be}(p))$ $ + d_{TV}({\\rm Be}(p), {\\rm Po}(p)) \n\\leq 2p^2$, which we apply with $p = x^j$. This completes the proof. \n\\hfill \\mbox{\\rule{0.5em}{0.5em}}\n\n\nThe random variable $X_{\\kappa}$ of Section \\ref{logsect} is the special case\n$c = 0$ of $X_{\\kappa,c}$. Further, for $c \\neq 0$,\n$$\n{\\Bbb E} X_{\\kappa,c} = \\kappa \\frac{1 - e^{-c}}{c}\n$$\nand\n$$\n{\\rm Var} X_{\\kappa,c} = \\kappa \\frac{1 - (1 + c)e^{-c}}{c^2}.\n$$\nThe density $g_c$ of $X_{\\kappa,c}$ may be found from the density $g$ of\n$X_{\\kappa}$ by observing that the log Laplace transforms, given by\n(\\ref{limitlt}) and (\\ref{limitlt1}), are related by\n$$\n\\psi_c(s) = \\frac{\\psi(c+s)}{\\psi(c)}\n$$\nso that\n$$\ng_c(z) = e^{-c z} g(z) \/ \\psi(c), \\ z \\geq 0.\n$$\nIn particular, from (\\ref{ignatov}),\n\\begin{equation}\\label{gcdensity}\ng_c(z) = \\frac{e^{- \\gamma \\kappa} e^{- c z} z^{\\kappa -\n1}}{\\Gamma(\\kappa) \\psi(c)},\\ 0 \\leq z \\leq 1.\n\\end{equation}\n\nFrom (\\ref{gcdensity}) the\nvalue of $c$ that maximizes the density\n$g_c(z)$ for fixed $z \\in [0,1]$ is the $c$ that maximizes\n$-c z - \\log \\psi(c)$, just as suggested by large deviation theory. \nThis $c$ is the solution of the equation\n$$\nc z = \\kappa ( 1 - e^{- c z}).\n$$\nUsing $z = 1$, we see from the heuristic (\\ref{local1}) that choosing\n$c$ to be the\n solution of $c= \\kappa (1-e^{-c})$ asymptotically maximizes ${\\Bbb P}(T_n = n)$; \nand from (\\ref{newmean}), this  also makes ${\\Bbb E} T_n \\sim n$.\n\nHowever, the heuristic in (\\ref{approx3}) and \n(\\ref{approx4}) suggests that better\napproximation should follow from choosing $c$ so that $g_c^{\\prime}(1-) =\n0$. From (\\ref{gcdensity}) and (\\ref{ignatov1}), we get \n\\begin{equation}\\label{ignatov3}\nc = \\frac{g^{\\prime}(1 -)}{g(1)} = \\kappa - 1.\n\\end{equation}\nFor this choice of $c$ we have $g_c^{\\prime}(1-) = 0$, and \n\\begin{equation}\\label{ignatov4}\n\\frac{g_c^{\\prime\\prime}(1 -)}{g_c(1)} = 1 - \\kappa.\n\\end{equation}\nA second order approximation in the spirit of Section \\ref{sect4} then leads \nus to the following heuristic: for any fixed $B$,\n\n {\\it In the case $\\kappa \\neq 1$}\n \\begin{equation}\\label{approx3c}\n d_{TV}(\\b C_B,\\b Z_B) \\bothsides \\frac{\\sigma_B^2}{n^2},\n \\end{equation}\n \n {\\it In the case $\\kappa = 1$}\n \\begin{equation}\\label{approx4c}\n d_{TV}(\\b C_B,\\b Z_B) = o\\left(\\frac{1}{n^2}\\right).\n \\end{equation}\n\nFor the case $B = [b] \\equiv \\{1,2,\\ldots,b\\}$, extensive numerical\ncomputations using the recurrence methods described in Section \\ref{sect9}\nsupport these conjectures for several of the combinatorial examples\ndiscussed earlier. In these cases, the bound in (\\ref{approx3c}) is of\norder $(b\/n)^2$. Finding the asymptotic form of this rate seems to be a\nmuch harder problem, since it seems to depend heavily on the value of\n$\\kappa$.\n\n\\subsection{Further examples}\\label{sect5.3}\n\nThe class of partitions of a set provides another example to show that \nthe choice of\n $x$ for good approximation is partly a matter of taste. In this example,\n $m_i \\equiv 1$, so that\n $$\n {\\Bbb E} T_n = \\sum_{i=1}^n \\frac{i m_i x^i}{i!} = x \\sum_{i=0}^{n-1}\n \\frac{x^i}{i!}.\n $$\n One choice of $x$ would be the exact solution $x^*$ of the equation ${\\Bbb E}\n T_n = n$, but this choice is poor since the definition of $x^*$ \nis complicated. \n A second choice which is more usable is to take $x = x^\\prime$, the \n solution of the equation $x e^x = n$. This is based on the\n observation that ${\\Bbb E} T_n \\sim x e^x$, provided $x = o(n)$.\n The solution  $x^\\prime$ has the form\n (cf. de Bruijn, 1981, p. 26)\n $$\n x^\\prime = \\log n - \\log \\log n + \\frac{\\log \\log n}{\\log n} + \\frac{1}{2}\n \\left( \\frac{\\log \\log n}{\\log n} \\right)^2 + O\\left( \\frac{\\log \\log\n n}{\\log^2 n} \\right).\n $$\n\n For set partitions,\n with either $x^*$ or $x^\\prime$ in the role of $x$, we have $\\sigma_n^2\n  \\sim x^2 e^x \\sim n \\log n$, and we can check that $| n - {\\Bbb E} T_n| =\n O(\\sqrt{n \\log n})$ is satisfied using $x=x^\\prime$. This corresponds\nto checking the condition in (\\ref{metathm1}). Comparing the condition\n${\\Bbb E} T_n \\sim n$ with the condition that $n - {\\Bbb E} T_n = O(\\sigma_n)$\nrequired by (\\ref{metathm1}), we see that in the logarithmic class the\nformer is too restrictive while for set partitions it is not\nrestrictive enough.\n\n\n\n\n \n \n\n\n \\section{Refining the combinatorial and independent\nprocesses}\\label{sect6}\n\n\\subsection{Refining and conditioning}\n\nAlthough the refinements considered in this section are complicated in\nnotation, the ingredients -- including geometric and Bernoulli random\nvariables and the counting formulas (\\ref{Rassembly}) -\n(\\ref{Rselection}) -- are\nsimpler than their unrefined counterparts.\n\nThe dependent random variables $C_i \\equiv C_i(n)$, which\n count the number of\ncomponents of weight $i$ in a randomly selected object of total\nweight $n$, may be refined as\n\\[\n  C_i = \\sum_{j = 1}^{m_i} D_{ij}.\n\\]\nHere we suppose that the $m_i$ possible structures\nof weight $i$ have been labelled $1,2,\\ldots,m_i$, and $D_{ij} \\equiv\nD_{ij}(n)$ counts the number of occurrences of the $j^{th}$ object of\nweight $i$.  The independent random variable $Z_i$ can also be refined, as\n\\[\n  Z_i = \\sum_{j=i }^{ m_i} Y_{ij},\n\\]\nwhere the $Y_{ij}$ are mutually independent, and for each $i$,\n$Y_{i1},Y_{i2},\\ldots,Y_{im_i}$ are identically distributed.  For\nassemblies, multisets, and selections respectively, the\ndistribution of $Y_{ij}$ is Poisson $( x^i\/i!)$ for $x>0$,\n geometric($x^i$) for $0<x<1$, or\nBernoulli($x^i\/(1+x^i))$ for $x>0$. If the choice of parameter $x$ is\ntaken as a function of $n$, then one can view $Y_{ij}$ as $Y_{ij}(n)$.\nFor assemblies, with $x > 0$,\n\\begin{equation}\\label{80a}\n{\\Bbb P}(Y_{ij}=k) = \\exp(-x^i\/i!) \\frac{(x^i\/i!)^k}{k!},\\ k=0,1,\\ldots.\n\\end{equation}\nFor multisets, with $0 < x < 1$, \n\\begin{equation}\\label{Zrefine}\n{\\Bbb P}(Y_{ij}=k) = (1-x^i)x^{ik},\\ k=0,1,\\ldots,\n\\end{equation}\nwhereas for selections, with $x > 0$, we have\n\\begin{equation}\\label{80c}\n{\\Bbb P}(Y_{ij}=k) = \n \\frac{1}{1+x^i} \\mbox{\\bf 1}(k=0) \\ + \\ \\frac{x^i}{1+x^i} \\mbox{\\bf 1}(k=1).\n\\end{equation}\n\nFor the full refined processes corresponding to a random object of size\n$n$ we denote the combinatorial process by\n\\[\n\\b D(n) \\equiv (D_{ij}(n),\\ 1 \\leq i \\leq n, 1 \\leq j \\leq m_i),\n\\]\nand the independent process by\n\\[\n\\b Y(n) \\equiv (Y_{ij},\\ 1 \\leq i \\leq n, 1 \\leq j \\leq m_i).\n\\]\nThe weighted sum $T_n = \\sum_1^n i Z_i$ is of course a\nweighted sum of the refined independent $Y$'s, since\n\\[\n  T_n = \\sum_{i=1}^n \\sum_{j=1}^{m_i} i Y_{ij}.\n\\]\n\\begin{theorem}\\label{refinedthm}\nFor assemblies, multisets, and selections, if ${\\Bbb P}(T_n=n)>0$, then \nthe refined combinatorial\nprocess, for a uniformly chosen object of weight $n$, is equal in\ndistribution to the independent process $\\b Y(n)$, conditioned on the\nevent $\\{ T_n=n \\}$, that is\n\\[ \\b D(n) \\stackrel{\\mbox{\\small d}}{=} (\\b Y(n)|T_n=n).  \\]\n\\end{theorem}\n\\noindent{\\bf Proof\\ \\ }\nJust as (\\ref{equaldist}) is a special case of Theorem\n\\ref{genequaldist} with $t=n$, so is this.  Imagine first the special case of\n(\\ref{equaldist}) with each $m_i \\equiv 1$, and then replicate $m_i$--\nfold the\nindex $i$ and its corresponding function $g_i$ and normalizing constant\n$c_i$.  The case $m_i=0$ for some $i$ is allowed. We have index set\n\\begin{equation}\n\\label{refineI}\n  I = \\{ \\alpha =(i,j): 1 \\leq i \\leq n, 1 \\leq j  \\leq m_i \\}\n\\end{equation}\nand weight function $w$ given by $ w(\\alpha) = i$ for  $\\alpha = (i,j) \\in I$.\n\nThe reader should be convinced by now, but for the record, here are the\ndetails.\nFor $\\b{b} \\equiv (b(\\alpha))_{\\alpha \\in I} \\in \\B{Z}_+^I$,\nwrite $\\b b \\cdot \\b w \\equiv \\sum_I w(\\alpha)b(\\alpha)$. Consider the number \n$R(n,\\b{b})$ of\nobjects of total weight $\\b b \\cdot \\b w =n$, having $b_\\alpha \\equiv b(\\alpha)$\n components of type $\\alpha$, for $\\alpha \\in I$.\n  For assemblies, the refined generalization  of Cauchy's formula is that\n\\begin{eqnarray}\nR(n, \\b{b}) & \\equiv &  | \\{ \\mbox{assemblies on [n]}: \\b{D} = \\b{b} \\} |\n\\nonumber \\\\\n& = &\n \\mbox{\\bf 1}(\\b b \\cdot \\b w =n) \n\\   n! \\ \\prod_{\\alpha \\in I} \\frac{1}{(i!)^{b(\\alpha) } \\ b(\\alpha) !},\n\\label{Rassembly}\n\\end{eqnarray}\nwhere $i =w(\\alpha)= $ the first coordinate of $\\alpha$. \nFor multisets, \n\\begin{eqnarray}\nR(n, \\b{b}) & \\equiv &  | \\{ \\mbox{multisets of weight n}: \\b{D} = \\b{b} \\} |\n \\\\ \\nonumber\n& = & \\mbox{\\bf 1}(\\b b \\cdot \\b w =n),\n\\label{Rmultiset}\n\\end{eqnarray}\nwhile for selections, \n\\begin{eqnarray}\nR(n, \\b{b}) & \\equiv &  | \\{ \\mbox{selections of weight n}: \\b{D} = \\b{b} \\} |\n\\\\  \\nonumber \n& = &\n \\mbox{\\bf 1}(\\b b \\cdot \\b w =n) \n\\    \\ \\prod_1^n {1  \\choose b_\\alpha }.\n\\label{Rselection}\n\\end{eqnarray}\n\nThese examples have the form\n\\begin{equation}\nR(n, \\b{b})  \\equiv   | \\{  \\b{D} = \\b{b} \\} | = \n \\mbox{\\bf 1}(\\b b \\cdot \\b w =n) \n\\  f(n)  \\ \\prod_{\\alpha \\in I} g_\\alpha(b_\\alpha),\n\\label{Rgeneral}\n\\end{equation}\nwith $f(n)=n!$ for assemblies and $f(n) \\equiv 1$ for multisets and\nselections. With $p(n)$ given by (\\ref{def p(n)}), we have the refined\nanalysis of the total number of structures of weight $n$:\n\\begin{equation}\n   p(n)= \\sum_{\\b b \\in \\B Z_+^I} R(n,\\b b).\n\\end{equation}\nPicking an object of weight $n$ uniformly defines the refined\ncombinatorial distribution\n\\begin{equation}\n  \\B P(\\b D(n) = \\b b) \\equiv   \\frac{R(n,\\b{b})}{p(n)} =\n \\mbox{\\bf 1}(\\b b \\cdot \\b w =n) \n\\  \\frac{f(n)}{p(n)}  \\ \\prod_I g_\\alpha(b_\\alpha).\n\\label{refinedcombdist}\n\\end{equation}\nObserve that with multisets, $g_\\alpha(k)=1$ for $ k \\in {\\Bbb Z}_+$;\nwith selections $g_\\alpha(k) = {1 \\choose k} = \\mbox{\\bf 1}(k=0 \\mbox{ or }1)$;\nand with assemblies, if $\\alpha =(i,j)$, then $g_\\alpha(k) =(1\/i!)^k\/k!,$\nfor $k \\in {\\Bbb Z}_+$. \nNow apply Theorem \\ref{genequaldist} with $\\mbox{\\boldmath $D$}_I$ in the role of $\\mbox{\\boldmath $C$}_I$,\n$Y_{ij} \\equiv Y_\\alpha$ in the role of $Z_\\alpha$, and $t=n$. \\hfill \\mbox{\\rule{0.5em}{0.5em}}\n\n{\\bf Remark}. It would be reasonable to consider (\\ref{Rassembly})\nthrough (\\ref{Rselection}) as the basic counting formulas, with\n(\\ref{Nassembly}) through (\\ref{Nselection}) as corollaries derived by\nsumming, and to consider the Poisson, geometric, and Bernoulli\ndistributions in (\\ref{Zrefine}) as the basic distributions, with the\nPoisson, negative binomial, and binomial distributions in\n(\\ref{Zassembly}) through (\\ref{Zselection}) derived by convolution. \n\n\\subsection{Total variation distance}\n\nSince the refined combinatorial process $\\b D(n)$ and the refined\nindependent process $\\b Y(n)$ are related by conditioning on the value\nof a weighted sum of the $Y$'s, Theorem \\ref{tvthm} applies. For $K\n\\subset I$, where $I$ is given by (\\ref{refineI}), write $\\b D_K$ and\n$\\b Y_K$ for our refined processes, restricted to indices in $K$. Write \n\\[\nR_K' \\equiv \\sum_{\\alpha \\in K} w(\\alpha) Y_\\alpha, \\ \\ \\ \\ \\\nS_K' \\equiv \\sum_{\\alpha \\in I-K} w(\\alpha) Y_\\alpha,\n\\]\nso that $T \\equiv T_n = R_K' + S_K'$. \n\n\\begin{theorem}\\label{tvrefine}\n\\begin{equation}\nd_{TV}(\\b D_K,\\b Y_K) \\ = d_{TV}((R_K'|T=n),R_K').\n\\label{refinedtveq,general}\n\\end{equation}\n\\end{theorem}\n\n\\noindent{\\bf Proof\\ \\ }\nThis is a special case of Theorem \\ref{tvthm}, with the\nindependent process $\\b Y(n) \\equiv \\b Y_I$ playing the role of $\\b Z_I$\nand $\\b D(n) \\equiv \\b D_I$ playing the role of $\\b C_I$.  \nTheorem  \n\\ref{refinedthm} \nis used to verify that the hypothesis \n(\\ref{tvhyp}) is satisfied, in the form $\\b D_I  \\stackrel{\\mbox{\\small d}}{=}\n(\\b Y_I|T=n)$.\n\\hfill \\mbox{\\rule{0.5em}{0.5em}}\n\nFor the special case where $B \\subset \\{1, \\ldots,n \\}$ and $K=\\{\\alpha\n= (i,j) \\in I: i \\in B \\}$,  denote the restriction of the refined\ncombinatorial process, restricted to sizes in $B$,\n by $\\b D_{B^*} \\equiv \\b D_K$,\nso that\n\\[\n\\b D_{B^*} \\equiv (D_{ij},\\ i \\in B, 1 \\leq j \\leq m_i),\n\\]\nand similarly define $\\b Y_{B^*}$.  In this special case, $R_K'\n= R_B \\equiv \\sum_{i \\in B} i Z_i$ is the weighted sum, restricted to B,\nfor the unrefined process, so (\\ref{refinedtveq,general}) reduces to\n\\begin{equation}\nd_{TV}(\\b D_{B^*},\\b Y_{B^*}) \\ = d_{TV}((R_B|T=n),R_B).\n\\label{refinedtveq}\n\\end{equation}\nFurthermore, by Theorem \\ref{tvthm} applied to the unrefined case, with\n$I=\\{1,\\ldots,n \\}$ and $w(i)=i$ , we see that  $ d_{TV}((R_B|T=n),R_B)$ \nis equal to\n $d_{TV}(\\b C_B,\\b Z_B)$.\n\nWe have here a most striking example of the situation analyzed in\nTheorem \\ref{tv=thm}, where taking functionals doesn't change a total\nvariation distance.  Namely, there is a functional $g: \\B Z_+^I\n\\rightarrow \\B Z_+^n$, which ``unrefines'', and the functional $h: \\B\nZ_+^B \\rightarrow \\B Z_+$ discussed in our second proof of Theorem\n\\ref{tvthm}, such that \n\\[\ng(\\b D_{B^*})=\\b C_B, \\ g(\\b Y_{B^*})=\\b Z_B, \\ \\ \\ \\\nh(\\b C_B) \\stackrel{\\mbox{\\small d}}{=} (R_B|T=n) , \\mbox{ and }h(\\b Z_B)=R_B,\n\\]\nso that, a priori via (\\ref{tvinequality}),\n\\begin{equation}\nd_{TV}(\\b D_{B^*},\\b Y_{B^*}) \\geq d_{TV}(\\b C_B,\\b Z_B) \\geq\nd_{TV}((R_B|T=n),R_B).\n\\label{surprise=}\n\\end{equation}\nPerhaps the result in (\\ref{refinedtveq}), which shows that equality\nholds throughout (\\ref{surprise=}), is surprising.\n\n\n\n\n\n\n\n \n \n\n\n \\section{Conditioning on events of moderate probability}\\label{sect7}\n\nWe consider random combinatorial structures conditioned on some event.\n Given that we have a good\napproximation by another process, this other\nprocess, conditioned on the same event, may yield a good approximation to\nthe conditioned combinatorial structure.  The conditioning event must\nhave moderate probability, large relative to the original approximation\nerror. In contrast, if the conditioning event is very unlikely then the \napproximating process must also be changed, as discussed in Section\n\\ref{sect8} on large deviations. \n\n\\subsection{Bounds for conditioned structures}\n\nIn this subsection, we consider bounds on total variation distance \nthat are inherited from an existing approximation, after additional\nconditioning is applied. \n\\begin{theorem}\\label{mild condition}\nLet $A \\subseteq B \\subseteq\n[n]$, and let  $h: \\B Z_+^B \\rightarrow \\{0,1\\}$ be measurable with\nrespect to coordinates in $A$. Let $\\b Z_B$, and $\\b C_B$ be arbitrary\nprocesses with values in $\\B Z_+^B$, and let $\\b Z_A$ and $\\b C_A $ denote\ntheir respective restrictions to coordinates in $A$. Let\n\\[\n\\b C_B^* \\stackrel{\\mbox{\\small d}}{=} (\\b C_B | h(\\b C_B) = 1),\n\\]\nand\n\\[\n\\b Z_B^* \\stackrel{\\mbox{\\small d}}{=} (\\b Z_B | h(\\b Z_B) = 1).\n\\]\nWrite\n$p  = \\B P(h(\\b Z_B) = 1),$\n$q  = \\B P(h(\\b C_B) = 1),$\n$d_B =  d_{TV}(\\b C_B, \\b Z_B)$, \n$d_A  =  d_{TV}(\\b C_A, \\b Z_A)$, and\nassume that $p>0$ and $q>0$. Then\n\\begin{eqnarray}\\label{tvbnd0}\nd_{TV}(\\b C_B^*, \\b Z_B^*) \n& \\leq & \\frac{1}{2} \\left| 1 - \\frac{q}{p} \\right| + \\frac{d_B}{p} \\\\\n& & \\nonumber \\\\\n& \\leq & \\frac{1}{p} \\left( \\frac{d_A}{2} +\nd_B \\right) \\label{tvbnd1} \\\\\n& & \\nonumber \\\\\n\\label{tvbnd2}\n& \\leq & \\frac{3}{2} \\frac{d_B}{p}.\n\\end{eqnarray}\n\n\\end{theorem}\n\\noindent{\\bf Proof\\ \\ }\n The second to last inequality follows from \nthe relation $| p - q | \\leq d_A,$\nand is useful when this is the extent of our ability to estimate\n$q$.\nThe last inequality follows simply from the fact that $d_A \\leq d_B$. To\nestablish the first inequality, we have\n\\begin{eqnarray*}\nd_{TV}(\\b C_B^*, \\b Z_B^*) & = & \\frac{1}{2} \\sum_{\\smb a \\in \\B Z_+^B} |\\B\nP(\\b C_B^* = \\b{a}) - \\B P(\\b Z_B^* = \\b{a}) |\\\\\n&&\\\\\n& = & \\frac{1}{2} \\sum_{\\smb{a}: h(\\smb{a}) = 1} \\left| \\frac{\\B P(\\b C_B\n= \\b{a})}{q} - \\frac{\\B P(\\b Z_B = \\b{a})}{p} \\right| \\\\\n&&\\\\\n& = & \\frac{1}{2} \\sum_{\\smb{a}: h(\\smb{a}) = 1} \\left| \\B P(\\b C_B\n= \\b{a}) \\left( \\frac{1}{q} - \\frac{1}{p}\\right) +\n\\frac{\\B P(\\b C_B\n= \\b{a}) - \\B P(\\b Z_B = \\b{a})}{p} \\right| \\\\\n&&\\\\\n& \\leq & \\frac{1}{2} \\left| \\frac{1}{q} - \\frac{1}{p} \\right| \n\\sum_{\\smb{a} : h(\\smb{a}) = 1}\\B P(\\b C_B = \\b{a}) \\\\\n&&\\\\\n& & \\ \\ \\ + \\frac{1}{2p} \n\\sum_{\\smb{a}: h(\\smb{a}) = 1} \\left| \\B P(\\b C_B\n= \\b{a}) - \\B P(\\b Z_B = \\b{a}) \\right| \\\\\n&&\\\\\n& = & \\frac{1}{2} \\left| \\frac{1}{q} - \\frac{1}{p} \\right| q\n+ \\frac{1}{2p} \n\\sum_{\\smb{a}: h(\\smb{a}) = 1} \\left| \\B P(\\b C_B\n= \\b{a}) - \\B P(\\b Z_B = \\b{a}) \\right| \\\\\n&&\\\\\n& \\leq & \\frac{1}{2} \\left| \\frac{1}{q} - \\frac{1}{p} \\right| q + \\frac{1}{2p} \n\\sum_{\\smb{a}} \\left| \\B P(\\b C_B\n= \\b{a}) - \\B P(\\b Z_B = \\b{a}) \\right| \\\\\n&&\\\\\n& = & \\frac{1}{2} \\left| 1 - \\frac{q}{p} \\right| + \\frac{d_B}{p}. \n\\end{eqnarray*}\n\\hfill \\mbox{\\rule{0.5em}{0.5em}}\n\n{\\bf Remark.} \nWhile the theorem above uses the notation $C_B$ and $Z_B$ to suggest\napplications where one process is obtained from an independent process by\nconditioning, no such structure is required.  An arbitrary discrete space\n$S$, together with an arbitrary functional $h:S \\rightarrow \\{0,1\\}$, may\nbe encoded in terms of $S=\\B Z_+^2$, with $A=\\{1\\}$ and $B=\\{1,2\\}$, so\nthat $h$ depends only on the first coordinate.  Thus Theorem \\ref{mild\ncondition} applies to\ndiscrete random objects in general.\n\n\\subsection{Examples}\n\n\\subsubsection{Random permutations}\n\nIn this case, the $Z_i$ are independent Poisson\ndistributed random variables, with $\\lambda_i \\equiv \\B E Z_i = 1\/i$.  \nIn {\\rm Arratia and Tavar\\'e (1992a)} \\ it is\nproved that for $1 \\leq b \\leq n$, the total variation distance $d_b(n)$\nbetween $(C_1(n),\\ldots,C_b(n))$ and $(Z_1,\\ldots,Z_b)$ satisfies $d_b(n)\n\\leq F(n\/b)$ where \n\\begin{eqnarray}\n  F(x) & \\equiv &\\sqrt{2 \\pi m} \\, \\frac{2^{m-1}}{(m-1)!} \\quad + \n\\frac{1}{m!} + 3 \\, \\left( \\displaystyle{\\frac{x}{e}}\\right)^{-x}, \n\\quad \\mbox{ with } m \\equiv \\lfloor x \n\\rfloor \\label{Fdef}\\\\\n& & \\nonumber\\\\\n& \\sim & \\left( \\frac{2e}{\\lfloor x-1 \\rfloor} \\right)^{\\lfloor x -1\n\\rfloor} \\nonumber\n\\end{eqnarray}\nas $x \\rightarrow \\infty$.\nTo get an approximation result for derangements, we use the functional $h$\nhaving $h((a_1,\\ldots,a_b))= \\mbox{\\bf 1}(a_1=0)$, with $A=\\{1\\}$ and\n$B=\\{1,2,\\ldots,b\\}$. This makes $\\b C_B^*$ the process counting cycles of\nsize at most $b$ in a randomly chosen derangement, and $\\b Z_B^*\n=(Z_1^*,Z_2^*,\\ldots,Z_b^*) \\stackrel{\\mbox{\\small d}}{=} (0,Z_2,\\ldots,Z_b)$. The total\nvariation distance $d_b^*(n)$ between $\\b C_B^*$ and $\\b Z_B^*$\nsatisfies $d_b^*(n) \\leq (3\/2)e \\ F(n\/b)$, simply by using\n(\\ref{tvbnd2}).  \n\nChanging random permutations to random \nderangements is a special case of conditioning on some fixed conditions of\nthe form $C_i(n) =c_i, i \\in A$, for given constants $c_i$, with $A \\subseteq\nB \\subseteq \\{1,2,\\ldots,b\\}$. In this situation, all the $Z_i^* $ are \nmutually independent, $Z_i^* \\equiv c_i$ for $i \\in A$,\nand  for $i \\notin A$,\\ \\ $Z_i^* \\stackrel{\\mbox{\\small d}}{=} Z_i$ is Poisson with mean $1\/i$.\n   Here, Theorem \\ref{mild condition}\nyields the bound $d_b^*(n) \\leq 3\/(2p) F(n\/b)$, where $p = \\B P(Z_i=c_i\n \\ \\forall i \\in A)$.  Theorem 3 in {\\rm Arratia and Tavar\\'e (1992a)}  \\ gives a different upper bound, namely\n$d_b^*(n) \\leq F((n-s)\/b) +2be ((n-s)\/(be))^{-(n-s)\/b}$, where $s= \\sum_{i\n\\in A} i c_i$.  Either of these two upper bounds may be smaller, depending\non the situation given by $A, b$, and the $c_i$.\n\nFor a more complicated conditioning in which the $Z_i^*$ are not mutually\nindependent, consider random permutations on $n$ objects conditional on\nhaving at least one cycle of length two or three. Here, $Z_2^*$ and $Z_3^*$\nare dependent, although the {\\it pair} $(Z_2^*,Z_3^*)$ and the variables\n$Z_1^*,Z_4^*,Z_5^*,\\ldots$ are mutually independent.  With $A=\\{2,3\\} \n\\subseteq B\n=\\{1,2,\\ldots,b\\}$, we have $p=\\B P(Z_2+Z_3 >0) = 1 - e^{-5\/6}$ and\n$d_b^*(n) \\leq 3\/(2p) F(n\/b)$.  Thus, for example with $b=3$, \nthe probability that a random permutation of $n$ objects \nis a derangement, given that\n$C_2(n)+C_3(n)>0$, can be approximated by $\\B P(Z_1^*=0)=1\/e$, with\nerror at most $3\/(2p) F(n\/3)$.  Similarly, the probability that a \nrandom permutation of $n$ objects \nhas a cycle of length 2, given that\n$C_2(n)+C_3(n)>0$, can be approximated by $\\B P(Z_2^*>0)=\\B\nP(Z_2>0|Z_2+Z_3>0) =(1-e^{-1\/2})\/(1-e^{-5\/6})$, with\nerror again at most $3\/(2p) F(n\/3)$. \n\nThe next example shows how to approximate easily the small component counts\nfor \n2--regular graphs by exploiting a decoupling result for the Ewens sampling\nformula with parameter $\\kappa = 1\/2$.\n\n\\subsubsection{2-regular graphs}\n\nThe combinatorial structure known as `2--regular graphs' is the assembly\nin which components are undirected cycles on three or more points, so\nthat\n\\begin{equation}\\label{2reg-mi}\nm_i = \\frac{1}{2} (i-1)!\\;\\mbox{\\bf 1}\\{i \\ge 3\\}.\n\\end{equation}\nLet $C_i^*(n)$ be the number of components of size $i$ in a random \n2--regular graph on $n$ points. A process that corresponds to this, with\nthe condition $\\mbox{\\bf 1}\\{i \\ge 3\\}$ removed, is the Ewens sampling formula with\nparameter $\\kappa = 1\/2$ described in Section\n\\ref{esfsect}. Observe that \n\\[\n\\b C^*(n) \\stackrel{\\mbox{\\small d}}{=} (\\b C(n) | C_1(n) = C_2(n) = 0).\n\\]\n\nThe bound \n\\[\nd_{TV}( (C_1,\\ldots,C_b), (Z_1,\\ldots,Z_b)) \\leq \\frac {2 b}{n}\n\\]\nis known from results of Arratia, Barbour and Tavar\\'e (1992). We are\ninterested in how this translates into a bound on\n\\[\nd_b^* \\equiv d_{TV}( (C_3^*,\\ldots,C_b^*), (Z_3,\\ldots,Z_b)).\n\\]\n\nWith $A = \\{1,2\\}, B = \\{1,2,\\ldots,b\\}$,\n$d_A \\leq 4\/n, d_B \\leq 2b\/n, p = \\B P(Z_1 = Z_2 = 0) = e^{-3\/4}$,\nthe inequality in (\\ref{tvbnd1}) guarantees that\n\\begin{eqnarray*}\nd_b^* & \\leq & \\frac{1}{p} \\left(\\frac{d_A}{2}  + d_B\\right) \\\\\n&&\\\\\n& \\leq & e^{3\/4} \\left( \\frac{2}{n} + \\frac{2 b}{n} \\right)\\\\\n&&\\\\\n& = & e^{3\/4} \\frac{2(b+1)}{n}. \n\\end{eqnarray*}\n\n\nFor an example that shows that the conditioning event can\nhave probability tending to zero, consider 2--regular graphs conditioned\non having no cycles of size less than or equal to $t \\equiv t(n) \\geq 2$. The\nprevious example is the special case $t = 2$. For $b > t$, we have\n\\[\n(C_{t+1}^*,\\ldots,C_b^*) \\stackrel{\\mbox{\\small d}}{=} (C_{t+1},\\ldots,C_b | C_1 = \\cdots =\nC_t = 0).\n\\]\nNow $d_A \\leq 2t\/n,\\ d_B \\leq 2b\/n$, and\n\\[\np = \\B P(Z_1= \\cdots = Z_t=0) = \\exp\\left(-\\frac{1}{2}(1+ \\cdots+\n1\/t)\\right)\n\\geq \\frac{1}{\\sqrt{e t}},\n\\]\nso (\\ref{tvbnd1}) establishes that \n\\begin{eqnarray*}\nd_b^* & \\leq & \\frac{1}{p}\\left(\\frac{d_A}{2} + d_B\\right)\\\\\n&&\\\\\n& \\leq & \\sqrt{e t} \\left(\\frac{t}{n} + \\frac{2b}{n}\\right).\n\\end{eqnarray*}\nThis provides a useful bound provided that $\\sqrt{t}b\/n$ is small. Note\nthat both $t$ and $b$ may grow with $n$, as long as $t \\leq b$.  \nFor example, conditional on no cycles of length less than or equal to $t\n= \\lfloor n^{2\/3-\\epsilon} \\rfloor$ this approximation successfully\ndescribes the distribution of the $k$ smallest cycles, for fixed $k$ as\n$n \\to \\infty$, by using $b = n^{2\/3}$. See Arratia and Tavar\\'e (1992b,\nTheorem 7) for related details.\n\n\n\n \n\n\n \\section{Large deviation theory}\\label{sect8}\n\\subsection{Biasing the combinatorial and independent processes}\n\nA guiding principle of large deviation theory is that unlikely events of\nthe form $\\{U \\geq u \\} $ or $\\{ U \\leq u \\} $ or $ \\{ U=u\\}$, where the\ntarget $u$ is far from $\\B E U$, can be studied by changing the measure\n$\\B P$ to another measure $\\B P_\\theta$ defined by\n\\begin{equation}\n\\frac{d \\B P_\\theta}{d \\B P} = \\frac{\\theta^U}{\\B E \\theta^U}.\n\\label{twist C}\n\\end{equation}\nObserve that for $\\theta=1$, the new measure $\\B P_\\theta$ coincides\nwith the original measure $\\B P$, regardless of the choice of $U$. \nThe parameter $\\theta$ is chosen so that the average\nvalue of $U$ under the new \nmeasure is $u$, i.e. $\\B E_\\theta U =u$. In  the literature on\nlarge deviations and statistical mechanics (cf. Ellis, 1985), the\nnotation is usually \n$\\theta \\equiv e^\\beta$, and our normalizing factor $\\B E \\theta^U$ is\nexpressed as the Laplace transform of the $\\B P$-distribution of \n$U$, parameterized by $\\beta$.\n\nFor the case of a combinatorial process $\\b C(n) =(C_1(n),\\ldots,C_n(n))$,\nwith  the total number of components \n$$\nK \\equiv K_n \\equiv C_1(n) + \\cdots + C_n(n)\n$$ \nin the role of $U$, this says to change from the measure $\\B P$, which\nmakes all possible structures equally likely, to the measure $\\B\nP_\\theta$, which selects a structure with bias proportional to\n$\\theta^{\\# {\\rm components }}$.  The Ewens sampling formula\ndiscussed in Section \\ref{esfsect} is exactly this in the case of random\npermutations, with $\\kappa$ playing the role of $\\theta$. This may \neasily be verified by comparing (\\ref{esfdef}) to Cauchy's formula, the\nspecial case $\\kappa = 1$ of (\\ref{esfdef}), in which the equality of\nnormalizing constants, with ${\\Bbb E} \\kappa^{K_n} = \\kappa_{(n)}$,\nexpresses a well known identity for Stirling numbers of the first kind.\n\nTheorem  \\ref{genequaldist} showed that many a combinatorial process is\nequal in distribution to a process of independent random variables,\nconditioned on the value of a weighted sum.  The next theorem asserts\nthat  this form is preserved\nby the change of measure from large deviation theory, provided that $U$ is\nalso a weighted sum. \n\nAs in the discussion before Theorem \\ref{genequaldist}, the weight\nfunction $\\mbox{\\boldmath $u$}$, just like the weight function $\\b w$, can take values in\n${\\Bbb R}$ or ${\\Bbb R}^d$.  In case the weights $\\mbox{\\boldmath $u$}$, and hence the random\nvariable $U$, takes values in ${\\Bbb R}^d$ with $d>1$, we take $\\theta >0$ to\nmean that $\\theta = (\\theta_1,\\ldots,\\theta_d) \\in (0,\\infty)^d$, and\nwith $U=(U_1,\\ldots,U_d), \\theta^U$ represents the product\n$\\theta_1^{U_1}\\cdots \\theta_d^{U_d}$.\n\n\\begin{theorem}\\label{twist dist thm}\nLet $I$ be a finite set, and for $\\alpha \\in I$, let $C_\\alpha$ and\n$Z_\\alpha$ be $\\B Z_+$-valued random variables.\nLet $\\b w = (w(\\alpha))_{\\alpha \\in\nI}$ and $\\b u = (u(\\alpha))_{\\alpha \\in\nI}$ be deterministic weight functions on $I$,\n with real values for $\\b u$, let $T = \\b w \\cdot \\b Z_I \\equiv \\sum_{\\alpha \\in\nI} w(\\alpha) Z_\\alpha$, and let $U= \\b u \\cdot \\b C_I$. Let $\\B P$ be a\nprobability measure and $t$ be a\n constant such that, under $\\B P$ the $Z_\\alpha$ are mutually\nindependent, \n$\\ \\B P(T=t) >0$, and $\\b C_I \\stackrel{\\mbox{\\small d}}{=} (\\b Z_I | T=t)$. Let $\\theta >0$\nbe any constant such that the random variable  $Y \\equiv\n \\theta^{\\b u \\cdot \\b Z_I}$ has ${\\Bbb E} Y\n < \\infty$. Let $\\B P_\\theta$,\nrestricted to the sigma--field generated by $\\b C_I$, be given by\n(\\ref{twist C}). Let ${\\Bbb P}_{\\theta}$, restricted to the sigma-field\ngenerated by $\\mbox{\\boldmath $Z$}_I$, be given by \n$$\n\\frac{d{\\Bbb P}_{\\theta}}{d{\\Bbb P}} = \\frac{Y}{{\\Bbb E} Y},\n$$\nso that the  $Z_\\alpha$ are  mutually independent under ${\\Bbb P}_{\\theta}$ with \n\\begin{equation}\n\\B P_\\theta(Z_\\alpha = k) = \\frac{\\theta^{u(\\alpha)k}}{\\B\nE\\theta^{u(\\alpha) Z_\\alpha}}\\,{\\Bbb P}(Z_{\\alpha} = k), k \\geq 0.\n\\label{twist Z marginal}\n\\end{equation}\nThen under $\\B P_\\theta, \\ \\ \\b C_I \\stackrel{\\mbox{\\small d}}{=} (\\b Z_I |T=t)$, that is\n\\begin{equation}\n\\B P_\\theta(\\b C_I = \\b a) = \\B P_\\theta(\\b Z_I = \\b a |T=t),\n\\end{equation}\nfor $\\b a \\in \\B Z_+^I$.\n\\end{theorem}\n\\noindent{\\bf Proof\\ \\ }\nFor $\\b a \\in \\B Z^I_+$,\n\\begin{eqnarray}\n\\B P_\\theta (\\b C_I = \\b a) &=& (\\B E \\theta^U)^{-1} \\ \\ \\theta^{\\b u \\cdot \\b a}\n \\ \\ \\B P(\\b C_I = \\b a)  \\nonumber \\\\ && \\nonumber \\\\\n&=& (\\B E \\theta^U)^{-1}\\ \\  \\theta^{\\b u \\cdot \\b a}\\ \\\n \\B P(\\b Z_I = \\b a| T=t)  \\nonumber \\\\ && \\nonumber \\\\\n&=& (\\B E \\theta^U)^{-1} \\ \\B P(T=t)^{-1} \\ \\theta^{\\b u \\cdot \\b a}\n \\ \\b 1(\\b w \\cdot \\b a = t) \\ \n\\B P(\\b Z_I = \\b a)\n\\label{stept1}\n\\end{eqnarray}\nNow\n\\[\n\\B P_\\theta(\\b Z_I=\\b a) = \\left(\\B E \\theta^{\\b u \\cdot \\b\nZ_I}\\right)^{-1} \\ \n\\theta^{\\b u \\cdot \\b a} \\ \\B P(\\b Z_I = \\b a)\n\\]\nso that \n\\begin{equation}\n\\B P_\\theta(\\b Z_I = \\b a |T=t) = \n \\left(\\B E \\theta^{\\b u \\cdot \\b Z_I}\\right)^{-1} \\ \n  \\ \\B P_\\theta(T=t)^{-1} \\ \\theta^{\\b u \\cdot \\b a}\n\\ \\b 1(\\b w \\cdot \\b a = t)  \\ \\B P(\\b Z_I = \\b a)\n\\label{stept2}.\n\\end{equation}\nComparing (\\ref{stept1}) and (\\ref{stept2}), we see both expressions are\nprobability densities on $\\B Z_+^I$ which are \nproportional to the same function of $\\b a$, and hence they are equal.\nFrom this it also follows that the normalizing constants are equal,\nwhich is written below with the combinatorial generating function on the\nleft, and the three factors determined by independent random variables\non the right:\n\\begin{equation}  \n\\B E \\theta^U  =\n \\B E \\theta^{\\b u \\cdot \\b Z_I}\\ \\frac{\\B P_\\theta(T=t)}{\\B P(T=t)}.\n\\label{free and sexy}\n\\end{equation}\n\\hfill \\mbox{\\rule{0.5em}{0.5em}}\n  \n\n  For\nthe case $U = K_n$, the total number of components,\n the $\\B P_\\theta$ measure corresponds to the following generalization\nof (\\ref{Zassembly}) through (\\ref{Zselection}).  For assemblies,\nmultisets, or selections, chosen with probability proportional to \n$\\theta^{\\# {\\rm components} }, \\b C(n) \\stackrel{\\mbox{\\small d}}{=} ((Z_1,...,Z_n)|Z_1+2 Z_2 +\\cdots\n+ n Z_n = n)$ where the $Z_i$ are mutually independent. With $\\theta, x\n> 0$, for assemblies we have\n\\begin{equation}\\label{twist Z dist}\n Z_i \\mbox{ is }  \\mbox{Poisson }\n(\\frac{m_i \\ \\theta \\  x^i}{i!}),\n\\end{equation}\nwhereas for multisets we require $x \\leq 1, \\theta x < 1$ and then\n$$\nZ_i \\mbox{ is negative binomial }(m_i,\\theta \\ x^i).\n$$\nFinally, for selections  \n$$\nZ_i \\mbox{ is binomial }(m_i, \\frac{\\theta \\  x^i}{1+\\theta x^i}).\n$$\nIn the general case, where $U = \\mbox{\\boldmath $u$} \\cdot \\mbox{\\boldmath $C$}(n)$ is a weighted sum of\ncomponent counts, so that \nthe selection bias is $\\theta^{\\b u \\cdot \\mbox{\\boldmath $C$}(n)}$, each factor \n$\\theta$ in (\\ref{twist Z dist}) above is replaced\nby $\\theta^{u(i)}$.  Furthermore, we observe that Theorems \\ref{tvthm},\n\\ref{refinedthm}, and\n \\ref{tvrefine} apply to $\\B P_\\theta$ in place of $\\B P$.  For the\nrefinements in Section \\ref{sect6}, for assemblies, multisets, and\nselections respectively, the distribution of $Y_{ij}$ is Poisson\n$(\\theta^{u(i)} \\ x^i\/i!)$, Geometric ($\\theta^{u(i)} \\ x^i$),\n or Bernoulli ($\\theta^{u(i)}\nx^i\/(1+\\theta^{u(i)} x^i))$.\n\nAn example where such a bias is well known is the random graph model\n${\\mathcal G}_{n,p}$; Bollob\\'as (1985). \nThis corresponds to picking a labelled graph on $n$\nvertices, where each of the potential edges is independently taken with\nprobability $p$; the unbiased case with all $2^{{n \\choose 2}}$ graphs\nequally likely is given by $p = 1\/2$. We need something like the refined\nsetup of Section 6 to be able to keep track of components in terms of\nthe number of edges in addition to the number of vertices. Using the\nfull refinement of Section 6, $D_{ij}$ counts the number of components\non $i$ vertices having the $j$th possible structure, for $j = 1, \\ldots,\nm_i$, in some fixed enumeration of these. The weight function should be\n$u(i,j)$ = \\# edges in the $j$th possible structure on $i$ vertices. With\n$\\theta = p\/(1-p)$, the ${\\Bbb P}_\\theta$ law of ${\\bf D}(n)$ is a\ndescription of ${\\mathcal G}_{n,p}$. A more natural refinement for this\nexample, intermediate between ${\\bf C}$ and ${\\bf D}$, would be the\nprocess ${\\bf A}$ with $A_{ik} = \\sum_{j:u(i,j) = k} D_{ij}$, the number\nof components with $i$ vertices and $k$ edges, for $k = i-1, \\ldots, {i\n\\choose 2}$. As in (96) and (97), the total variation distances are\ninsensitive to the amount of refining. Presumably there are interesting\nresults about random graphs that could easily be deduced from estimates\nof the total variation distance in Theorem 5.\n\nOne form of the general large deviation heuristic is that \nfor a process $\\b C$, conditioned on the event $\\{ U \\geq u \\} $ \nwhere  $U$ is a functional of the process and $u > \\B EU$, the ${\\Bbb P}-$law\nof the conditioned process\nis nicely approximated by the ${\\Bbb P}_\\theta-$ law of $\\b C$, where\n$\\theta$ is chosen so that $\\B E_\\theta U=u$.  We are interested in \n the special case where the\nfunctional $U$ is a weighted sum, and the distribution of $\\b C$ \nunder $\\B P$\nis that of an independent process $\\b Z$ conditioned on the value of\nanother weighted sum $T$.  In this case,\n Theorem \\ref{tvthm} yields a direct quantitative\nhandle on the quality of approximation by the $\\B P_\\theta$-distribution\nof the independent process, provided we condition on \nthe event $\\{ U= u \\} $ instead of the event $\\{ U \\geq u \\} $.\n\n\\begin{theorem}\\label{twist tvthm} \nAssume the hypotheses and notation of Theorems \\ref{tvthm} and\n \\ref{twist dist thm} combined.\nFor $B \\subset I$ write $U_B \\equiv \\sum_{\\alpha \\in B} u(\\alpha)\nZ_\\alpha$, so that $U_I \\equiv \\b u \\cdot \\b Z_I$.  \nWrite ${\\mathcal L}_\\theta$ for distributions governed by $\\B\nP_\\theta$, so that the conclusion of Theorem \\ref{twist dist thm} may be\nwritten\n\\[\n{\\mathcal L}_\\theta (\\b C_I) = {\\mathcal L}_\\theta (\\b Z_I | T=t),\n\\]\nand Theorem \\ref{tvthm} states that for $B \\subset I$\n\\begin{equation}\nd_{TV} ({\\mathcal L}_{\\theta}(\\b C_B),{\\mathcal L}_\\theta (\\b Z_B) \\ ) =\nd_{TV}({\\mathcal L}_\\theta (R_B|T=t), {\\mathcal L}_\\theta (R_B) \\ ).\n\\end{equation}\n Assume that $u$ is such that $\\B P(U=u)>0$. Then under the further\nconditioning on $U=u$, \n$$\nd_{TV} ({\\mathcal L}_1(\\b C_B|U=u),{\\mathcal L}_\\theta (\\b Z_B) \\ ) = \\ \\ \\ \\ \\\n\\ \\ \\ \n$$\n\\begin{equation}\n\\ \\ \\ \\ \\ \\\nd_{TV}({\\mathcal L}_\\theta ((U_B,R_B)|U_I=u,T=t), {\\mathcal L}_\\theta ((U_B,R_B))).\n\\label{twisty}\n\\end{equation}\n\\end{theorem}\n\\noindent{\\bf Proof\\ \\ }\nObserve first that \n\\begin{equation}\n\\label{theta id}\n{\\mathcal L}_1(\\b C_I|U=u) = {\\mathcal L}_\\theta(\\b C_I|U=u),\n\\end{equation}\nso that it suffices to prove (\\ref{twisty}) with the subscript\n$\\theta$ appearing on all four distributions, i.e.\n$$\nd_{TV} ({\\mathcal L}_\\theta(\\b C_B|U=u),{\\mathcal L}_\\theta (\\b Z_B) \\ ) =\\ \\ \\\n\\ \\ \\ \n$$\n\\begin{equation}\\label{twisty2}\n\\ \\ \\ \\ \\ \\\nd_{TV}({\\mathcal L}_\\theta((U_B,R_B)|U_I=u,T=t), {\\mathcal L}_\\theta ((U_B,R_B))).\n\\end{equation}\nObserve next that this is a special case of Theorem \\ref{tvthm}, but\nwith two--component weights $w^*(\\alpha) \\equiv (u(\\alpha),w(\\alpha))$\nin the role of $w(\\alpha)$. \n For example, in the usual combinatorial\ncase, with $I =[n]$ and $w(i)=i$, and further specialized to $U=K_n=$\nthe total number of components, so that $u(i)=1$, we have that $\\b w^*$\ntakes values in $\\B R^2$, with $w^*(i)=(1,i)$. \\hfill \\mbox{\\rule{0.5em}{0.5em}}\n\n\\medskip\n\\noindent{\\bf Discussion.}  The proof of the previous theorem helps make it clear\nthat the free parameter $x$, such that ${\\mathcal L}((Z_1,\\ldots,Z_n)|T_n=n)$\ndoes not vary with $x$, is analogous to the parameter $\\theta$, such\nthat relation (\\ref{theta id}) holds.  With this perspective, the\ndiscussion of an appropriate choice of $x$ in Section \\ref{sect4} and\nSection \\ref{sect5.2} is\nsimply giving details in some special cases of the general large deviation\nheuristic.  Note that $T_n$ is a sufficient statistic for $x$, while $U$\nis a sufficient statistic for $\\theta$.\n\n\n\n\n\n\nThere are three distributions involved in the discussion above: the first\nis ${\\mathcal L}(\\b C_I|U=u)$, corresponding to a combinatorial distribution\nconditioned on the value of \na weighted sum $U$, the second is ${\\mathcal L}_\\theta(\\b C_I)$,\nwhich is a biased version of the combinatorial distribution, and the \nthird is ${\\mathcal\nL}_\\theta(\\b Z_I)$, which governs an independent process.  Theorem \\ref{tvthm}, used\nwith Theorem \\ref{twist dist thm}, compares the second and third of\nthese; Theorem \\ref{twist tvthm}\nabove compares the first and third of these; and the following theorem\ncompletes the triangle, by comparing the first and second distributions.\n\n\\begin{theorem}\\label{leg3}\nIn the setup of Theorem \\ref{twist tvthm},  for $B \\subset I$,\n\\begin{equation}\nd_{TV} ({\\mathcal L}_1(\\b C_B|U=u),{\\mathcal L}_\\theta (\\b C_B) \\ ) =\n\\label{leg3eq}\n\\end{equation}\n\\[ d_{TV}({\\mathcal L}_\\theta ((U_B,R_B)|U_I=u,T=t), {\\mathcal L}_\\theta\n((U_B,R_B)|T=t \\ )).\n\\]\n\\end{theorem}\n\\noindent{\\bf Proof\\ \\ }\nBy Theorem \\ref{twist dist thm}, together with (\\ref{theta id}),\n the left side of\n(\\ref{leg3eq}) is equal to $ d_{TV} ({\\mathcal L}_\\theta(\\b Z_B|U_I=u,T=t),\n{\\mathcal L}_\\theta (\\b Z_B|T=t) \\ ) $. We modify the second proof of Theorem\n\\ref{tvthm} as follows:  replace $\\B P$  by $\\B P_\\theta$, use two--component \nweights, replace the original\nconditioning $T=t$ by $U_I=u$, and then  further condition\n on $\\{T=t \\}$. Explicitly, the functional $h$ on $ \\B Z_+^B$\ndefined by $h(\\b a)= \\sum_{\\alpha \\in B}\na(\\alpha)(u(\\alpha),w(\\alpha))$ is a sufficient statistic, and the sign\nof \n$\\B P_\\theta(\\b Z_B= \\b a|U_I=u,T=t) - \\B P_\\theta(\\b Z_B=\\b a | T = t) $ \nis equal to the sign of  $\\B P_\\theta((U_B,R_B)=h(\\b a)|U_I=u,T=t)- \n\\B P_\\theta((U_B,R_B)=h(\\b a)|T=t)$, i.e. the sign depends on $\\b a$\nonly through the value of $h (\\b a)$.\n \\hfill \\mbox{\\rule{0.5em}{0.5em}}\n\nObserve that Theorem \\ref{twist tvthm} \ncontains Theorem \\ref{tvthm} as a special case, by\ntaking weights $u(\\alpha) \\equiv 0$ and target $u=0$, so that $\\B\nP_\\theta = \\B P$ and the extra conditioning event $\\{ U=u \\}$ has\nprobability one.  \n\n\\subsection{Heuristics for good approximation of conditioned\ncombinatorial structures}\n\nThe following applies to weighted sums $U$ in general, but to be concrete\nwe present the special case $U=K_n$.  \nLet $K \\equiv K_n$ be the total number of components of some assembly, multiset,\nor selection of total weight $n$, and let some deterministic target $k \\equiv\nk(n)$ be given.  The goal is to describe an independent process to \napproximate $\\b C(n)$, conditioned on\nthe event $\\{K \\geq k \\}$, in case $k$ is large compared to $\\B EK$; or\nconditioned on \nthe event $\\{K \\leq k \\}$, in the opposite case; or more simply,\nconditioned on the event $\\{ K=k \\}$. We accomplish this by picking the\nfree parameters $\\theta$ and $x$ in (\\ref{twist Z dist}) so that\nsimultaneously $\\B E(Z_1+\\cdots +Z_n)$ is close to $k$ and $\\B ET_n$ is\nclose to $n$.  \n\nFor example, to study random permutations on $n$ objects, conditional on\nhaving at least $5 \\log n$ cycles, or conditional on having exactly\n$\\lfloor 5 \\log n \\rfloor$ cycles,  or conditional on having at most $0.3\n\\log n $ cycles, we propose using $x=1$, and $\\theta = 5$ or $0.3$.  The\nindependent process with this choice of parameter should be a good\napproximation for both the conditioned random permutations and for the\nEwens sampling formula.  As a corollary, the Ewens sampling formula\nshould be a good approximation for the conditioned permutations; see\nArratia, Barbour and Tavar\\'e (1994).\n\nFor assemblies, multisets and selections in the logarithmic class\ndiscussed in Section \\ref{logsect}, in which ${\\Bbb E} Z_i \\sim\n\\kappa\/i$, biasing by $\\theta^K$ yields ${\\Bbb E}_{\\theta} Z_i \\sim\n\\kappa \\theta\/i$, so that the Ewens sampling formula with\nparameter $\\kappa \\theta$ is a useful approximation for the biased\nmeasures. In particular, the heuristics (\\ref{approx3b}) and\n(\\ref{approx4b}) should apply in the following form: for fixed \n$B \\subseteq [n]$\n\n {\\it In the case $\\kappa \\theta \\neq 1$}\n \\begin{equation}\\label{approx3new}\n d_{TV}({\\mathcal L}_{\\theta}(\\mbox{\\boldmath $C$}_B),{\\mathcal L}_{\\theta}(\\mbox{\\boldmath $Z$}_B)) \\sim \n\\frac{1}{2} |\\kappa \\theta - 1|\n \\frac{{\\Bbb E}_{\\theta}|R_B - {\\Bbb E}_\\theta R_B|}{n},\n \\end{equation}\n \n {\\it In the case $\\kappa \\theta = 1$}\n \\begin{equation}\\label{approx4new}\n d_{TV}({\\mathcal L}_{\\theta}(\\mbox{\\boldmath $C$}_B),{\\mathcal L}_{\\theta}(\\mbox{\\boldmath $Z$}_B)) = \no\\left(\\frac{1}{n}\\right).\n \\end{equation}\n\nFor random permutations, for which $\\kappa = 1$, with $B = \\{1,2,\\ldots,b\\}$\nthe bound\n$$\nd_{TV}({\\mathcal L}_{\\theta}(\\mbox{\\boldmath $C$}_B),{\\mathcal L}_{\\theta}(\\mbox{\\boldmath $Z$}_B)) \\leq c(\\theta)\n\\frac{b}{n}\n$$\nwas established via a particular coupling in Arratia, Barbour and\nTavar\\'e (1992), and the asymptotic relation (\\ref{approx3new}) has\nbeen established by Arratia, Stark and Tavar\\'e (1994).\n\n\nTo show how the parameters $x$ and $\\theta$ may interact, we consider\nrandom permutations with $k(n)$ further away from $\\log n$. Assume that\n$k(n)$ is given such that as $n \\rightarrow \\infty$,\n\\[\n  k\/ \\log n \\rightarrow \\infty, \\ \\ \\ \\ k\/n \\rightarrow 0.\n\\]\nThen we would take\n\\begin{equation}\n\\theta \\equiv \\theta(n) = \\frac{k}{\\log(n\/k)}, \\ \\ \\ \\ \\ x \\equiv x(n) =\ne^{-\\theta \/n}.\n\\label{ewen pair}\n\\end{equation}\nObserve that $\\theta\/n \\rightarrow 0$, so that $x \\rightarrow 1$ and $1-\nx \\sim \\theta\/n$, and\n$\\theta \\rightarrow \\infty$, so that $x^n = \\exp(-\\theta) \\rightarrow 0$.\nHence\n\\[ \n\\B E T_n = \\theta \\sum_1^n x^i \\sim \\theta \\sum_0^\\infty x^i =\n\\theta \\frac{1}{1-x} \\sim n\n\\]\nand \n\\[\n\\B E K_n = \\theta \\sum_1^n \\frac{x^i}{i} \\sim -\\theta \\log(1-x) \\sim\n\\theta \\log (\\frac{n}{\\theta}) \\sim k.\n\\]\nWith this choice of parameters $\\theta$ and $x$ the independent Poisson\nprocess $(Z_1,Z_2,\\ldots)$ should be a good approximation for random\npermutations, conditioned either on having exactly $k$ cycles, or on\nhaving at least $k$ cycles.\n \n\n\n\n\n \\section{The generating function connection and moments}\\label{sect9}\n\n\nIn this section, we relate the probabilistic technique to the more\nconventional one based on generating functions; Wilf (1990). \nOne reason for this is to \nprovide a simple method, based on an idea of Shepp and Lloyd (1966), for \ncalculating moments of component counts for combinatorial structures. A \nsecond reason is to provide a framework within which detailed estimates and \nbounds for total variation distances can be obtained  by using  the results \nof Theorems \\ref{tvthm} and \\ref{twist tvthm}, together with analytic \ntechniques such as Darboux's method or the transfer methods of\nFlajolet and Odlyzko (1990).\n\n  Throughout, we let \n$p(n,k)$ be the number of objects of weight $n$ having $k$ components,\nso that\n$p(n) = \\sum_{k=1}^n p(n,k)$ is the number of objects of weight \n$n$. Finally, recall that $m_i$ is the number of available structures\nfor a component of size $i$.\n\n\\subsection{Assemblies}\n\nWe form the exponential generating functions\n\\begin{equation}\\label{egf3}\n\\hat P(s,\\theta) \\equiv 1 + \\sum_{n=1}^{\\infty} \\left(\\sum_{k=1}^n p(n,k) \\theta^k\n \\right) \\frac{s^n}{n!},\n\\end{equation}\n\\begin{equation}\\label{egf1}\n\\hat P(s) \\equiv 1 + \\sum_{n=1}^{\\infty} p(n) \\frac{s^n}{n!} = \\hat\nP(s,1),\n\\end{equation}\nand\n\\begin{equation}\\label{egf2}\n\\hat M(s) \\equiv  \\sum_{n=1}^{\\infty} m_n \\frac{s^n}{n!}.\n\\end{equation}\nFor assemblies, (\\ref{Nassembly}) gives\n$$\np(n,k) = \\sum_{\\mbox{\\boldmath $a$}} N(n,\\mbox{\\boldmath $a$}) = \\sum_{\\b a} n! \\prod_{j=1}^n\n\\left(\\frac{m_j}{j!}\\right)^{a_j} \\frac{1}{a_j!},\n$$\nwhere $\\sum_{\\b a}$ is over $\\{\\b a \\in \\B Z_+^n: \\sum i a_i = n, \\sum\na_i = k\\}$. It follows that\n\\begin{eqnarray}\n\\hat P(s,\\theta) & = & 1 + \\sum_{n=1}^\\infty \\sum_{k=1}^n \\sum_{\\b a}\n\\prod_{j=1}^n \\left( \\frac{\\theta m_j s^j}{j!}\\right)^{a_j}\n\\frac{1}{a_j!} \\nonumber \\\\\n&&\\nonumber \\\\\n& = & \\prod_{j=1}^{\\infty} \\exp\\left(\\frac{\\theta m_j s^j}{j!}\\right)\n\\nonumber \\\\\n&& \\nonumber \\\\\n& = & \\exp \\left( \\theta \\hat M(s) \\right) \\label{egf4}.\n\\end{eqnarray}\nEquation (\\ref{egf4}) is the well-known exponential generating function\nrelation for assemblies (cf. Foata, 1974), which has as a special case the \nrelationship\n\\begin{equation}\\label{egf5}\n\\hat P(s) = \\exp\\left( \\hat M(s) \\right).\n\\end{equation}\n\nRecall from Section 8 that in studying  large deviations  of $K_n$,\nthe number of components in the structure of total weight $n$, we were led \nto the measure ${\\Bbb P}_\\theta$\ncorresponding to sampling with probability proportional to $\\theta^{K_n}$. \nIt follows from (\\ref{Nassembly}) that there is a normalizing constant \n$p_\\theta(n)$ such that\n\\begin{eqnarray*}\np_\\theta(n)\\,\\B P_{\\theta}(\\b C(n) = \\b a) & = &\n \\theta^{a_1+ \\cdots+a_n} N(n,\\mbox{\\boldmath $a$}) \\\\\n& = & n! x^{-n} \\prod_{j=1}^n \n\\left(\\frac{\\theta m_j x^j}{j!}\\right)^{a_j} \\frac{1}{a_j!}\\ \\mbox{\\bf 1}\\left(\n\\sum_{l=1}^n l a_l = n \\right)\n\\end{eqnarray*}\nfor any $x > 0$. Clearly, \n\\begin{eqnarray}\np_{\\theta}(n) & = &  \\sum_{k=1}^n p(n,k) \\theta^k \\nonumber \\\\\n & = & n!  [s^n] \\hat P(s, \\theta) \\label{normconst1}\\\\\n & = & p(n) \\B E(\\theta^{K_n}) \\label{normconst2},\n\\end{eqnarray}\nwhere $\\B E \\equiv {\\Bbb E}_1$ denotes expectation with respect to the uniform measure ${\\Bbb P} \\equiv\n{\\Bbb P}_1$, corresponding to $\\theta = 1$. \n\nNext we explore the connection with the\nprobability generating function (pgf) of the random variable $T_n \\equiv\n\\sum_{j=1}^n j Z_j$, where the $Z_j$ are independent Poisson distributed\nrandom variables with mean\n$$\n{\\Bbb E}_{\\theta} Z_j \\equiv \\theta \\lambda_j = \\theta \\frac{m_j x^j}{j!}.\n$$\nRecall  that the pgf of a Poisson-distributed random variable $Z$ with\nmean $\\lambda$ is\n$$\n{\\Bbb E}_{\\theta} s^Z \\equiv \\sum_{j=0}^\\infty {\\Bbb P}_{\\theta}(Z=j) s^j \\, = \n\\exp(-\\lambda (1-s)),\n$$\nso using the independence of the $Z_j$,\n\\begin{eqnarray*}\n{\\Bbb E}_{\\theta} s^{T_n} & = & {\\Bbb E}_{\\theta} s^{\\sum_{j=1}^n j Z_j} \\\\\n&&\\\\\n& = & \\prod_{j=1}^n {\\Bbb E}_{\\theta} \\left(s^j\\right)^{Z_j}\\\\\n&&\\\\\n& = & \\exp\\left( - \\theta \\sum_{j=1}^n \\lambda_j (1 - s^j) \\right)\n\\end{eqnarray*}\nThus\n\\begin{eqnarray*}\n{\\Bbb P}_{\\theta}(T_n = n) & = & [s^n]\\, {\\Bbb E}_{\\theta} s^{T_n} \\\\\n&&\\\\\n& = & \\exp\\left( -\\theta \\sum_{j=1}^n \\lambda_j \\right)\\,[s^n] \\exp\n\\left(\\theta \\sum_{j=1}^n \\lambda_j s^j\\right)\\\\\n&&\\\\\n& = & \\exp\\left( -\\theta \\sum_{j=1}^n \\lambda_j \\right)\\,[s^n] \\exp\n\\left(\\theta \\sum_{j=1}^\\infty \\lambda_j s^j\\right)\\\\\n&&\\\\\n& = & \\exp\\left( -\\theta \\sum_{j=1}^n \\lambda_j \\right)\\,[s^n] \\exp\\left( \\theta \\hat M(sx)\\right)\\\\\n&&\\\\\n& = & \\exp\\left( -\\theta \\sum_{j=1}^n \\lambda_j \\right)\\,[s^n] \\hat P(sx, \\theta),\n\\end{eqnarray*}\nusing (\\ref{egf4}) at the last step. Thus, via  (\\ref{normconst1}),\n\\begin{equation}\\label{egf6}\n{\\Bbb P}_{\\theta}(T_n = n) = \\exp\\left(- \\theta \\sum_{j=1}^n \\lambda_j\\right) \n\\frac{x^n p_{\\theta}(n)}{n!},\n\\end{equation}\nas can also be calculated from (\\ref{assprobT=t}) and (\\ref{free and sexy}) \nfor the special case $U = K_n$.\n\n\nThe next result gives a simple expression for the joint moments \nof the component counts. We use the notation $y_{[n]}$ to denote the falling \nfactorial $y(y-1)\\cdots(y-n+1)$.\n\\begin{lemma}\\label{moments ass}\nFor $(r_1,\\ldots, r_b) \\in {\\Bbb Z}_+^b$ with $m = r_1 + 2 r_2 +\n \\cdots + b r_b$, we have\n\\begin{equation}\\label{meanass}\n{\\Bbb E}_{\\theta} \\prod_{j=1}^b (C_j(n))_{[r_j]} = \\mbox{\\bf 1}(m \\leq n)\\,x^{-m} \n\\frac{n!}{p_{\\theta}(n)} \\frac{p_{\\theta}(n-m)}{(n-m)!} \\prod_{j=1}^b \n\\left( \\frac{\\theta m_j x^j}{j!}\\right)^{r_j}.\n\\end{equation}\n\\end{lemma}\n\n\\noindent{\\bf Proof\\ \\ } The key step is the substitution of $a_1,\\ldots,a_b$ for $a_1-r_1,\\ldots,a_b - r_b$ \nin the third equality below. For $m \\leq n$, we have\n\\begin{eqnarray*}\n{\\Bbb E}_{\\theta} \\prod_{j=1}^b (C_j(n))_{[r_j]} & = & \\sum_{a_j \\geq r_j,\nj=1,\\ldots,b}\n\\sum_{a_{b+1},\\ldots,a_n: \\sum j a_j = n} (a_1)_{[r_1]}\\cdots (a_b)_{[r_b]}\n\\frac{n!}{x^n p_{\\theta}(n)}\\\\ \n& & \\ \\ \\times \\prod_{j=1}^n \\left( \\frac{\\theta m_j\nx^j}{j!}\\right)^{a_j} \\frac{1}{a_j!} \\\\\n&&\\\\\n& = & \\frac{n!}{x^n p_{\\theta}(n)} \\prod_{j=1}^b \\left( \\frac{\\theta m_j\nx^j}{j!}\\right)^{r_j}\\,\\sum \\sum \\prod_{j=1}^b \\left( \\frac{\\theta m_j\nx^j}{j!}\\right)^{a_j - r_j}\\\\ \n& & \\ \\ \\times \\frac{1}{(a_j - r_j)!} \\prod_{j=b+1}^n \n\\left( \\frac{\\theta m_j x^j}{j!}\\right)^{a_j} \\frac{1}{a_j!} \\\\\n&&\\\\\n& = & \\frac{n!}{x^n p_{\\theta}(n)} \\prod_{j=1}^b \\left( \\frac{\\theta m_j\nx^j}{j!}\\right)^{r_j}\\,\\sum_{a_1,\\ldots,a_{n}: \\sum j a_j = n-m} \\\\\n&&\\\\\n& & \\ \\ \\times \\prod_{j=1}^n \n\\left( \\frac{\\theta m_j x^j}{j!}\\right)^{a_j} \\frac{1}{a_j!} \\\\\n&&\\\\\n& = & \\frac{n!}{x^n p_{\\theta}(n)} \\prod_{j=1}^b \\left( \\frac{\\theta m_j\nx^j}{j!}\\right)^{r_j}\\, \\frac{x^{n-m} p_{\\theta}(n-m)}{(n-m)!}.\n\\end{eqnarray*}\n\\hfill \\mbox{\\rule{0.5em}{0.5em}}\n\n\n\\noindent{\\bf Remark: } If $\\{Z_i\\}$ are  mutually independent Poisson random \nvariables with ${\\Bbb E}_\\theta Z_i = \\theta m_i x^i \/ i!$, then the product term on the  right of equation (\\ref{meanass}) is precisely \n${\\Bbb E}_{\\theta} \\prod_{j=1}^b (Z_j)_{[r_j]}$.\n\n\\noindent{\\bf Remark: } In the special case of permutations, in which $m_i =\n(i-1)!$ and $p(n) = n!$, the normalizing constant $p_{\\theta}(n)$  is\ngiven by $p_{\\theta}(n) = \\theta(\\theta+1)\\cdots(\\theta+n-1)$, and\nequation (\\ref{meanass}) reduces to \n$$\n{\\Bbb E}_{\\theta} \\prod_{j=1}^b (C_j(n))_{[r_j]} = \\mbox{\\bf 1}(m \\leq n)\\, \n{{\\theta+n-m-1} \\choose {n-m}}{{\\theta+n-1} \\choose {n}}^{-1}\n\\prod_{j=1}^b \\left( \\frac{\\theta}{j}\\right)^{r_j},\n$$\na result of Watterson (1974).\n\n\\subsection{Multisets}\n\n\nFor multisets, the (ordinary) generating functions are\n\\begin{equation}\\label{ogf3}\nP(s,\\theta) \\equiv 1 + \\sum_{n=1}^{\\infty} \\left(\\sum_{k=1}^n p(n,k) \\theta^k\n \\right) s^n,\n\\end{equation}\n\\begin{equation}\\label{ogf1}\nP(s) \\equiv 1 + \\sum_{n=1}^{\\infty} p(n) s^n = P(s,1),\n\\end{equation}\nand\n\\begin{equation}\\label{ogf2}\nM(s) \\equiv  \\sum_{n=1}^{\\infty} m_n s^n.\n\\end{equation}\nIn this case, using (\\ref{Nmultiset}) gives\n$$\np(n,k) = \\sum_{\\mbox{\\boldmath $a$}} N(n,\\mbox{\\boldmath $a$}) = \\sum_{\\b a} \\prod_{j=1}^n\n{{m_j+a_j-1} \\choose {a_j}},\n$$\nthe sum $\\sum_{\\b a}$ being over $\\{\\b a \\in \\B Z_+^n: \\sum i a_i = n, \\sum\na_i = k\\}$. It follows that\n\\begin{eqnarray}\nP(s,\\theta) & = & 1 + \\sum_{n=1}^\\infty \\sum_{k=1}^n \\sum_{\\b a}\n\\prod_{i=1}^n {{m_i+a_i-1} \\choose {a_i}} (\\theta s^i)^{a_i} \\nonumber \\\\\n&&\\nonumber \\\\\n& = & \\prod_{i=1}^{\\infty} (1- \\theta s^i)^{-m_i}\\label{ogf7}\\\\\n&& \\nonumber\\\\\n& = & \\exp\\left(- \\sum_{i=1}^{\\infty} m_i \\log(1 - \\theta s^i)\\right)\n\\nonumber\\\\\n&&\\nonumber \\\\\n& = & \\exp\\left(\n\\sum_{i=1}^{\\infty} m_i \\sum_{j=1}^{\\infty} \\frac{ (\\theta\ns^i)^j}{j}\\right)\\nonumber \\\\\n&&\\nonumber \\\\\n& = & \\exp\\left(\n\\sum_{j=1}^{\\infty}\\frac{\\theta^j}{j} \\sum_{i=1}^{\\infty} m_i  s^{ij}\\right)\n\\nonumber \\\\\n&& \\nonumber \\\\\n& = & \\exp\\left(\n\\sum_{j=1}^{\\infty}\\frac{\\theta^j}{j} M(s^j)\\right). \\label{ogf9}\n\\end{eqnarray}\nSee Foata (1974) and Flajolet and Soria (1990) for example.\n\nUnder the measure $\\B P_{\\theta}$, there is a normalizing constant $p_{\\theta}(n)$ such that \n\\begin{eqnarray*}\np_{\\theta}(n)\\,\\B P_{\\theta}(\\b C(n) = \\b a) & = & \n\\prod_{i=1}^n {{m_i +\na_i -1} \\choose {a_i}} \\theta^{a_i}\\,\\mbox{\\bf 1}\\left(\\sum_{l=1}^n l a_l =\nn\\right) \\\\\n& = & x^{-n} \\prod_{l=1}^n (1 - \\theta x^l)^{-m_l}  \n\\prod_{i=1}^n {{m_i +\na_i -1} \\choose {a_i}} (1-\\theta x^i)^{m_i} (\\theta x^i)^{a_i}\\\\\n& & \\ \\ \\times \n\\mbox{\\bf 1}\\left(\\sum_{l=1}^n l a_l = n\\right),\n\\end{eqnarray*}\nfor any $0 < x < 1$. Indeed,\n\\begin{equation}\\label{ogf11}\np_{\\theta}(n) = p(n) \\B E_1(\\theta^{K_n}) = [s^n] P(s,\\theta),\n\\end{equation}\nwhere $p_{\\theta}(0) \\equiv 1.$\n\n In this case, the relevant $Z_j$ are independent negative binomial random variables with parameters $m_i$ and $\\theta x^i$ and pgf\n$$\n{\\Bbb E}_{\\theta} s^{Z_i} = \\left(\\frac{1-\\theta x^i}{1-\\theta x^is}\\right)^{m_i}.\n$$\nUsing the independence of the $Z_j$ once more, the pgf of \n$T_n$ may be found as\n\\begin{eqnarray}\n{\\Bbb E}_{\\theta} s^{T_n} & = & \\prod_{i=1}^n {\\Bbb E}_{\\theta} \\left(s^i\\right)^{Z_i}\n\\nonumber \\\\\n&& \\nonumber \\\\\n& = & \\prod_{i=1}^n \\left(\\frac{1-\\theta x^i}{1-\\theta\n(xs)^i}\\right)^{m_i}\\nonumber \\\\\n&& \\nonumber \\\\\n& = & \\left(\\prod_{i=1}^n (1-\\theta x^i)^{m_i}\\right) \n\\prod_{i=1}^n (1-\\theta (xs)^i)^{-m_i} \\label{ogf13}.\n\\end{eqnarray}\nUsing (\\ref{ogf7}), we see that\n\\begin{eqnarray*}\n{\\Bbb P}_{\\theta}(T_n = n) & = & [s^n]\\, {\\Bbb E}_{\\theta} s^{T_n} \\\\\n&&\\\\\n& = & \\left(\\prod_{i=1}^n (1-\\theta x^i)^{m_i}\\right) [s^n] \\exp\\left(-\n\\sum_{i=1}^n m_i \\log(1 - \\theta (xs)^i)\\right)\\\\\n&&\\\\\n& = & \\left(\\prod_{i=1}^n (1-\\theta x^i)^{m_i}\\right) [s^n] \\exp\\left(-\n\\sum_{i=1}^{\\infty} m_i \\log(1 - \\theta (xs)^i)\\right)\\\\\n&&\\\\\n& = & \\left(\\prod_{i=1}^n (1-\\theta x^i)^{m_i}\\right) [s^n]\nP(sx,\\theta),\n\\end{eqnarray*}\nso that\n\\begin{equation}\\label{ogf12}\n{\\Bbb P}_{\\theta}(T_n = n) = \\left(\\prod_{i=1}^n (1-\\theta x^i)^{m_i}\\right) \nx^n p_{\\theta}(n).\n\\end{equation}\nEquation (\\ref{ogf12}) can also be calculated from (\\ref{assprobT=t}) \nand (\\ref{free and sexy}) for the special case $U = K_n$.\n\nIn order to calculate moments of the component counts $\\b C(n)$, it is\nconvenient to use a variant on a theme of Shepp and Lloyd (1966). We\nassume that $M(s)$ has positive radius of convergence, $R$.\nAs above, let $Z_1,Z_2,\\ldots$ be mutually independent negative binomial random\nvariables, $Z_i$ having parameters $m_i$ and $\\theta x^i$, where $0 < x\n< \\min\\{R,1,\\theta^{-1}\\}$. Let $T_\\infty \\equiv \\sum_{i=1}^{\\infty} i Z_i$. \nNote that $T_{\\infty}$ is almost surely finite, because\n$$\n\\B E_{\\theta} T_{\\infty} = \\sum_{i=1}^{\\infty} \\frac{i m_i \\theta x^i}{1\n- \\theta x^i} \\leq \\frac{\\theta x}{1 - \\theta x} M^{\\prime}(x) < \\infty.\n$$\nThe distribution of $T_{\\infty}$ follows from (\\ref{ogf7}), (\\ref{ogf13}) and \nmonotone convergence since \n$$\n{\\Bbb E}_{\\theta} s^{T_{\\infty}} = \\frac{P(sx,\\theta)}{P(x,\\theta)}.\n$$\nHence\n\\begin{equation}\\label{ogf14}\n{\\Bbb P}_{\\theta}(T_{\\infty} = n) = x^n p_{\\theta}(n) \/ P(x,\\theta),\nn=0,1,\\ldots.\n\\end{equation}\nFurther, for $\\b a \\in {\\Bbb Z}_+^n$ and  $\\b Z(n) \\equiv (Z_1,\\ldots,Z_n)$\n\\begin{equation}\\label{ogf15}\n{\\Bbb P}_{\\theta}(\\b C(n) = \\b a) = {\\Bbb P}_{\\theta}(\\b Z(n) = \\b a | T_{\\infty} =\nn).\n\\end{equation}\nThis follows from the statement (\\ref{twist Z dist}) that\n$$\n{\\Bbb P}_{\\theta}(\\mbox{\\boldmath $C$}(n) = \\mbox{\\boldmath $a$}) = {\\Bbb P}_{\\theta}(\\mbox{\\boldmath $Z$}(n) = \\mbox{\\boldmath $a$} | T_n = n),\n$$\nand the observation that\n\\begin{eqnarray*}\n{\\Bbb P}_{\\theta}(\\mbox{\\boldmath $Z$}(n) = \\mbox{\\boldmath $a$} | T_n = n) & = & \\frac{{\\Bbb P}_{\\theta}(\\mbox{\\boldmath $Z$}(n) = \n\\mbox{\\boldmath $a$}, T_n = n)}{{\\Bbb P}_{\\theta}(T_n = n)} \\\\\n& = & \\frac{{\\Bbb P}_{\\theta}(\\mbox{\\boldmath $Z$}(n) = \\mbox{\\boldmath $a$}, T_n = n) {\\Bbb P}_{\\theta}(Z_{n+1} = \nZ_{n+2} = \\cdots = 0)}{ {\\Bbb P}_{\\theta}(T_n = n) {\\Bbb P}_{\\theta}(Z_{n+1} = Z_{n+2} = \\cdots = 0)} \\\\\n& = & \\frac{{\\Bbb P}_{\\theta}(\\mbox{\\boldmath $Z$}(n) = \\mbox{\\boldmath $a$}, T_{\\infty} = n)}{{\\Bbb P}_{\\theta}\n(T_{\\infty} = n)} \\\\\n& = & {\\Bbb P}_{\\theta}(\\mbox{\\boldmath $Z$}(n) = \\mbox{\\boldmath $a$} | T_{\\infty} = n).\n\\end{eqnarray*}\n\nNow let $\\Phi: {\\Bbb Z}_+^{\\infty} \\to {\\Bbb R}$, and set $\\mbox{\\boldmath $C$}_n \\equiv\n(C_1(n),\\ldots,C_n(n),0,0,\\ldots)$ with $\\mbox{\\boldmath $C$}_0 \\equiv (0,0,\\ldots)$. \nThe aim is to find an easy way to\ncompute ${\\Bbb E}_{\\theta}^n( \\Phi) = {\\Bbb E}_{\\theta}\\Phi(\\mbox{\\boldmath $C$}_n)$.\nIt is convenient to use the notation ${\\Bbb E}_{x,\\theta}$ to denote\nexpectations computed under the independent negative binomial measure\nwith parameters $x$ and $\\theta$. Shepp and Lloyd's method in the\npresent context is the observation, based on (\\ref{ogf14}) and\n(\\ref{ogf15}), that ${\\Bbb E}_{x,\\theta}(\\Phi | T_{\\infty} = n) = {\\Bbb E}_{\\theta}^n(\\Phi)$, so that\n\\begin{eqnarray}\n{\\Bbb E}_{x,\\theta} (\\Phi) & = & \\sum_{n=0}^\\infty {\\Bbb E}_{x,\\theta}( \\Phi |\nT_{\\infty} = n) {\\Bbb P}_{\\theta}(T_{\\infty} = n) \\nonumber \\\\\n&& \\nonumber \\\\\n& = & \\sum_{n=0}^\\infty {\\Bbb E}^n_{\\theta}( \\Phi)\nx^n p_{\\theta}(n)\/ P(x,\\theta). \\label{sandl1}\n\\end{eqnarray}\nThis leads to the result that\n\\begin{equation}\\label{ogf16}\n{\\Bbb E}^n_{\\theta}(\\Phi) = \\frac{[x^n] {\\Bbb E}_{x,\\theta}(\\Phi)\nP(x,\\theta)}{p_{\\theta}(n)}.\n\\end{equation}\n\nFor $r \\geq 1, jr \\leq n$, we use this method to calculate\nthe falling factorial moments \n${\\Bbb E}_{\\theta} (C_j(n))_{[r]}$.  This  determines all moments, \nsince $C_j(n)_{[r]} \\equiv 0$ if $jr > n$.\nIn this case $\\Phi(x_1,x_2,\\ldots) = (x_j)_{[r]}$, and\n\\begin{eqnarray*}\n{\\Bbb E}_{x,\\theta}(\\Phi) & = & {\\Bbb E}_{x,\\theta} (Z_j)_{[r]} \\\\\n& = & \\frac{\\Gamma(m_j+r)}{\\Gamma(m_j)} \\left( \\frac{\\theta x^j}{1 -\n\\theta x^j}\\right)^r.\n\\end{eqnarray*}\nHence we have\n\\begin{eqnarray}\n{\\Bbb E}_{\\theta}^n(\\Phi) & = & \\frac{\\Gamma(m_j+r)}{p_{\\theta}(n)\n\\Gamma(m_j)} [x^n] P(x,\\theta) \\left( \\frac{\\theta x^j}{1 -\n\\theta x^j}\\right)^r \\nonumber\\\\\n&&\\nonumber\\\\\n& = & \\frac{\\theta^r \\Gamma(m_j+r)}{p_{\\theta}(n) \\Gamma(m_j)} [x^{n-\nrj}] P(x,\\theta) \\sum_{l=0}^{\\infty} {{r+l-1} \\choose {l}} \\theta^l\nx^{jl} \\nonumber\\\\\n&& \\nonumber\\\\\n& = & \\frac{\\theta^r \\Gamma(m_j+r)}{p_{\\theta}(n) \\Gamma(m_j)}\n\\sum_{l=0}^{\\lfloor n\/j \\rfloor- r} {{r+l-1} \\choose {l}} \\theta^l\np_{\\theta}(n - jr - jl) \\nonumber \\\\\n&& \\nonumber\\\\\n& = & \\frac{\\Gamma(m_j+r)}{p_{\\theta}(n) \\Gamma(m_j)}\n\\sum_{m = r}^{\\lfloor n\/j \\rfloor} {{m-1} \\choose {r-1}} \\theta^m\np_{\\theta}(n - jm).\n\\end{eqnarray}\n\n\\noindent{\\bf Remark: } See Hansen (1993) for related material. The Shepp and \nLloyd method can also be used in\nthe context of assemblies, for which (\\ref{ogf14}) holds with\n\\begin{equation}\\label{ogf26}\n{\\Bbb P}_{\\theta}(T_{\\infty} = n) = \\frac{x^n}{n!} p_{\\theta}(n) \/ \\hat P(x,\\theta),\nn=0,1,\\ldots\\ .\n\\end{equation}\nThis provides another proof of Lemma \\ref{moments ass}.\nSee Hansen (1989) for the case of random mappings, and Hansen (1990) for\nthe case of the Ewens sampling formula.\n\n\\subsection{Selections}\n\nThe details for the case of selections are similar to those for\nmultisets. Most follow by replacing $\\theta$ and $m_i$ by $-\\theta$ and\n$-m_i$ respectively in the formulas for multisets. First, we have from (\\ref{Nselection})\n$$\np(n,k) = \\sum_{\\mbox{\\boldmath $a$}} N(n, \\mbox{\\boldmath $a$}) = \\sum_{\\b a} \\prod_{j=1}^n\n{{m_j} \\choose {a_j}},\n$$\nthe sum $\\sum_{\\b a}$ being over $\\{\\b a \\in \\B Z_+^n: \\sum i a_i = n, \\sum\na_i = k\\}$. Therefore \n\\begin{eqnarray}\nP(s,\\theta) & = & 1 + \\sum_{n=1}^\\infty \\sum_{k=1}^n \\sum_{\\b a}\n\\prod_{i=1}^n {{m_i} \\choose {a_i}} (\\theta s^i)^{a_i} \\nonumber \\\\\n&&\\nonumber \\\\\n& = & \\prod_{i=1}^{\\infty} (1 + \\theta s^i)^{m_i}\\label{sgf7}\\\\\n&& \\nonumber\\\\\n& = & \\exp\\left(\n\\sum_{j=1}^{\\infty}\\frac{(-1)^{j-1} \\theta^j}{j} M(s^j)\\right), \\label{sgf9}\n\\end{eqnarray}\nthe last following just as (\\ref{ogf9}) followed from (\\ref{ogf7}).\n See Foata (1974), Flajolet and \nSoria (1990) for example.\n\nUnder the measure $\\B P_{\\theta}$, there is a normalizing constant $p_\\theta(n)$  such that \n\\begin{eqnarray*}\np_{\\theta}(n)\\,\\B P(\\mbox{\\boldmath $C$}(n) = \\b a) & = & \\prod_{i=1}^n {{m_i}\n\\choose {a_i}} \\theta^{a_i}\\,\\mbox{\\bf 1}\\left(\\sum_{l=1}^n l a_l =\nn\\right) \\\\\n& = & x^{-n} \\prod_{l=1}^n (1 + \\theta x^l)^{m_l}  \n\\prod_{i=1}^n {{m_i} \\choose {a_i}} (1+ \\theta x^i)^{-m_i} (\\theta x^i)^{a_i}\\\\\n& & \\ \\ \\times \n\\mbox{\\bf 1}\\left(\\sum_{l=1}^n l a_l = n\\right),\n\\end{eqnarray*}\nfor any $ x>0$;\n$p_{\\theta}(n)$ is given in (\\ref{ogf11}) once more.\n In this case, the $Z_j$ are independent \nbinomial random variables with pgf\n\\begin{equation}\\label{sgf20}\n{\\Bbb E}_{\\theta} s^{Z_i} = \\left(\\frac{1+ \\theta x^i s}{1+ \\theta x^i}\\right)^{m_i},\n\\end{equation}\nand the pgf of $T_n$ is\n\\begin{eqnarray}\n{\\Bbb E}_{\\theta} s^{T_n} & = & \n\\left(\\prod_{i=1}^n (1+\\theta x^i)^{-m_i}\\right) \n\\prod_{i=1}^n (1+\\theta (xs)^i)^{m_i} \\label{sgf13}.\n\\end{eqnarray}\nIt follows from (\\ref{ogf7}) that\n\\begin{eqnarray*}\n{\\Bbb P}_{\\theta}(T_n = n) & = & \n\\left(\\prod_{i=1}^n (1+ \\theta x^i)^{- m_i}\\right) [s^n]\nP(sx,\\theta),\n\\end{eqnarray*}\nso that\n\\begin{equation}\\label{sgf12}\n{\\Bbb P}_{\\theta}(T_n = n) = \\left(\\prod_{i=1}^n (1+\\theta x^i)^{-m_i}\\right) \nx^n p_{\\theta}(n).\n\\end{equation}\n\n\nThe joint moments of the counts can be calculated using the Shepp and\nLloyd construction once more. In particular, equations (\\ref{ogf14}) and\n(\\ref{ogf15}) hold, and we can apply (\\ref{ogf16}) with\n${\\Bbb E}_{x,\\theta}(\\Phi)$ denoting expectation with respect to independent\nbinomial random variables $Z_1,Z_2,\\ldots$ with distribution determined\nby (\\ref{sgf20}). \n\nAs an example, we use this method to calculate\n${\\Bbb E}_{\\theta} (C_j(n))_{[r]}$ for $r \\geq 1, jr \\leq n$. Since\n\\begin{eqnarray*}\n{\\Bbb E}_{x,\\theta}(\\Phi) & = & {\\Bbb E}_{x,\\theta} (Z_j)_{[r]} \\\\\n& = & (m_j)_{[r]} \\left( \\frac{\\theta x^j}{1 + \\theta x^j}\\right)^r,\n\\end{eqnarray*}\nfrom (\\ref{ogf16}) we have\n\\begin{eqnarray}\n{\\Bbb E}_{\\theta}^n(\\Phi) & = & \\frac{(m_j)_{[r]}}{p_{\\theta}(n)}\n[x^n] P(x,\\theta) \\left( \\frac{\\theta x^j}{1 +\n\\theta x^j}\\right)^r \\nonumber\\\\\n&&\\nonumber\\\\\n& = & \\frac{(m_j)_{[r]}}{p_{\\theta}(n)}\n\\sum_{m = r}^{\\lfloor n\/j \\rfloor} {{m-1} \\choose {r-1}} (-1)^{m-r} \\theta^m\np_{\\theta}(n - jm).\n\\end{eqnarray}\n\n\\subsection{Recurrence relations and numerical methods}\n\n\nWe saw in Theorems \\ref{tvthm} and \\ref{twist tvthm} that for \nany $B \\subseteq [n]$, the total\nvariation distance between $\\mbox{\\boldmath $C$}_B$ and $\\mbox{\\boldmath $Z$}_B$ can be expressed in terms\nof the distributions of random variables $S_B$ and $R_B$ defined by\n\\begin{equation}\\label{egf8}\nS_B = \\sum_{i \\in [n] - B} i Z_i,\n\\end{equation}\nand\n\\begin{equation}\\label{egf7}\nR_B = \\sum_{i \\in B} i Z_i \\equiv S_{[n]-B}.\n\\end{equation}\nSpecifically,\n\\begin{eqnarray}\nd_{TV}({\\mathcal L}_{\\theta}(\\mbox{\\boldmath $C$}_B), {\\mathcal L}_{\\theta}(\\b Z_B)) & = &\n\\frac{1}{2} \\B P_{\\theta}(R_B>n) \\label{tvcalc2} \\\\\n& &\\ \\  + \\frac{1}{2} \\sum_{r=0}^n \\B P_{\\theta}( R_B=r) \n \\left| \\frac{\\B P_{\\theta}( S_B=n-r)}{\\B P_{\\theta}(T_n=n)} - 1 \\right|. \n\\nonumber\n\\end{eqnarray}\n\nA direct attack on estimation of $d_{TV}({\\mathcal L}_{\\theta}(\\mbox{\\boldmath $C$}_B), {\\mathcal L}_{\\theta}(\\mbox{\\boldmath $Z$}_B))$  can be based\non a generating function approach to the asymptotics (for large $n$) of\nthe terms in (\\ref{tvcalc2}). In the setting of assemblies, this uses\nthe result before (\\ref{egf6})\nfor ${\\Bbb P}_{\\theta}(T_n = n)$, and the fact that for $k \\geq 0$\n\\begin{eqnarray}\n{\\Bbb P}_{\\theta}(S_B = n-k) & = & [s^{n-k}]\\,\\exp \\left(- \\theta \\sum_{i \\in [n]-B}\n\\lambda_i(1 - s^i)\\right) \\nonumber\\\\\n&&\\nonumber\\\\\n& = & \\exp\\left(- \\theta \\sum_{i \\in [n]-B} \\lambda_i\\right)\\,[s^{n-k}]\n\\exp \\left( \\theta \\sum_{i \\in [n]-B} \\lambda_i s^i + \\theta \n\\sum_{i > n} \\lambda_i s^i\\right) \\nonumber \\\\\n&&\\nonumber\\\\\n& = & \\exp\\left(- \\theta \\sum_{i \\in [n]-B} \\lambda_i\\right)\\,[s^{n-k}]\n\\hat P(sx,\\theta)\\, \\exp \\left( - \\theta \\sum_{i \\in B} \\lambda_i\ns^i\\right). \\nonumber\\\\\n&&\\label{recur0}\n\\end{eqnarray}\nFor applications of this technique, see Arratia, Stark and Tavar\\'e\n(1994), and Stark (1994b).\n\nIt is also useful to have a recursive method for calculating the\ndistribution of $R_B$ for any $B \\subseteq [n]$. For assemblies,\n\\begin{equation}\\label{recur1}\n{\\Bbb E}_{\\theta} s^{R_B} = \\exp\\left(- \\sum_{i \\in B} \\theta \\lambda_i\n\\right)\\, \\exp\n\\left( \\sum_{i \\in B} \\theta \\lambda_i s^i\\right).\n\\end{equation}\nWrite \n$$\nG_B(s) = \\sum_{i \\in B} \\theta \\lambda_i,\n$$\nand\n$$\nF_B(s) = \\exp G_B(s) \\equiv \\sum_{k=0}^\\infty q_B(k) s^k,\n$$\nwith $q_B(0) \\equiv 1$. Differentiating with respect to $s$ shows that $s\nF^\\prime_B(s) = s G^\\prime_B(s)\\,F_B(s)$ (cf. Pourahmadi, 1984), and\nequating coefficients of $s^k$ gives \n$$\nk q_B(k) = \\sum_{i=1}^k g_B(i) q_B(k-i),\\ k=1,2,\\ldots\n$$\nwhere\n\\begin{equation}\\label{recur2}\ng_B(i) = \\theta i \\lambda_i \\,\\mbox{\\bf 1}(i \\in B).\n\\end{equation}\nSince $p_B(k) \\equiv {\\Bbb P}_{\\theta}(R_B = k) = \\exp(- G_B(1)) q_B(k)$,\nwe find that \n\\begin{equation}\\label{recur3}\nk p_B(k) = \\sum_{i=1}^k g_B(i) p_B(k-i),\\ k=1,2,\\ldots\n\\end{equation}\nwith $p_B(0) = \\exp(-G_B(1))$.\nThe relation (\\ref{recur3}) has been exploited in the case of uniform\nrandom permutations ($\\theta = 1, \\lambda_i = 1\/i$) by Arratia and Tavar\\'e\n(1992a).\n\nFor multisets, the analog of (\\ref{recur0}) is\n\\begin{eqnarray}\n{\\Bbb P}_{\\theta}(S_B = n-k) & = & \\left(\\prod_{i \\in [n]-B} (1- \\theta \nx^i)^{m_i}\\right) \n[s^{n-k}] \\prod_{i \\in [n]-B} (1-\\theta (xs)^i)^{-m_i}\\nonumber\\\\\n&&\\nonumber\\\\\n& = & \\left(\\prod_{i \\in [n]-B} (1-\\theta x^i)^{m_i}\\right) [s^{n-k}]\n\\prod_{i \\in [n]-B} (1-\\theta (xs)^i)^{-m_i} \\nonumber\\\\\n&&\\nonumber\\\\\n& & \\ \\ \\times  \\prod_{i>n} (1-\\theta (xs)^i)^{-m_i}\\nonumber\\\\\n&&\\nonumber\\\\\n& = & \\left(\\prod_{i \\in [n]-B} (1- \\theta x^i)^{m_i}\\right) [s^{n-k}]\nP(sx, \\theta) \\prod_{i \\in B} (1-\\theta (xs)^i)^{m_i}.\\nonumber\\\\\n&&\\label{recur4}\n\\end{eqnarray}\n\nTo develop a recursion for $p_B(k) \\equiv {\\Bbb P}_{\\theta}(R_B = k)$, we can\nuse logarithmic differentiation; cf Apostol (1976), Theorem 14.8.\nFirst, we have\n\\begin{equation}\\label{recur5}\n{\\Bbb E} s^{R_B} = \\prod_{i \\in B} (1 - \\theta  x^i)^{m_i}\\,\n\\prod_{i \\in B} (1 -  \\theta  x^i s^i)^{- m_i}.\n\\end{equation}\nDefine \n$$\nG_B(s) = \\sum_{i \\in B} m_i s^i,\n$$\nand\n$$\nF_B(s) = \\prod_{i \\in B} (1 -  \\theta  x^i s^i)^{- m_i} \n\\equiv \\sum_{k=0}^\\infty q_B(k) s^k,\n$$\nwith $q_B(0) = 1$. Then\n$$\n\\log F_B(s) = \\sum_{j=1}^{\\infty} \\frac{\\theta^j}{j} G_B( (xs)^j).\n$$\nDifferentiating with respect to $s$ and simplifying shows that \n$$\ns F^\\prime_B(s) = \\left( \\sum_{i \\geq 1} g_B(i) s^i \\right)\\,F_B(s),\n$$\nwhere\n\\begin{equation}\\label{recur6}\ng_B(i) = x^i \\sum_{k|i} k m_k \\theta^{i\/k}\\,\\mbox{\\bf 1}(k \\in B).\n\\end{equation}\nEquating coefficients of $s^k$ gives \n$$\nk q_B(k) = \\sum_{i=1}^k g_B(i) q_B(k-i),\\ k=1,2,\\ldots.\n$$\nSince $p_B(k) \\equiv {\\Bbb P}_{\\theta}(R_B = k) = \\prod_{i \\in B} (1 - \\theta\nx^i)^{m_i}\\, q_B(k)$,\nit follows that \n\\begin{equation}\\label{recur7}\nk p_B(k) = \\sum_{i=1}^k g_B(i) p_B(k-i),\\ k=1,2,\\ldots\n\\end{equation}\nwith $p_B(0) = \\prod_{i \\in B} (1 - \\theta x^i)^{m_i}$.\n\nFor selections, we have the following identity, valid for $k \\geq 0$:\n$$\n{\\Bbb P}_{\\theta}(S_B = n-k)  = \n\\left(\\prod_{i \\in [n]-B} (1+ \\theta x^i)^{-m_i}\\right) [s^{n-k}]\nP(sx, \\theta) \\prod_{i \\in B} (1+\\theta (xs)^i)^{-m_i}.\n$$\nIf we define $p_B(k) \\equiv {\\Bbb P}_\\theta(R_B =k)$, then from equation\n(\\ref{recur7}) we obtain \n\\begin{equation}\\label{stnew}\nk p_B(k) = \\sum_{i=1}^k g_B(i) p_B(k-i),\\ k=1,2,\\ldots\n\\end{equation}\nwhere \n$$\ng_B(i) = - x^i \\sum_{k|i} k m_k (- \\theta)^{i\/k} \\mbox{\\bf 1}(k \\in B),\n$$\nand \n$$\np_B(0) = \\prod_{i \\in B} (1 + \\theta x^i)^{- m_i}.\n$$\n\n\n\n \n \n\n\n \\section{Proofs by overpowering the conditioning}\\label{sect10}\n\nThe basic strategy for making the relation $\\b C_I \\stackrel{\\mbox{\\small d}}{=} (\\b Z_I|T=t)$\ninto a useful approximation is to pick the free parameter $x$ in the\ndistribution of $\\mbox{\\boldmath $Z$}_I$ so that the conditioning is not severe, i.e. so\nthat $\\B P(T=t)$ is not too small.  It is sometimes possible to get useful upper\nbounds on events involving the combinatorial process $\\b C_I$ by\ncombining upper bounds on the probability of the same event for the\nindependent process, together with lower bounds for $\\B P(T=t)$.\nThe formal description of this strategy is given by the following lemma.\n\n\\begin{lemma}\\label{dumbdumb}\nAssume that  $\\b C_I \\stackrel{\\mbox{\\small d}}{=} (\\b Z_I|T=t)$\nand that $h$ is a nonnegative functional of these processes, i.e.\n\\[\nh: \\B Z_+^I \\rightarrow \\B R_+.\n\\]\nThen\n\\begin{equation}\n\\label{dumb but useful}\n\\B E h(\\b C_I) \\leq \\frac{\\B E h(\\b Z_I)}{\\B P(T=t)}.\n\\end{equation}\n\\end{lemma}\n\\noindent{\\bf Proof\\ \\ }\n\\[\n\\B E h(\\b C_I) =\\frac{\\B E(h(\\b Z_I) \\mbox{\\bf 1} (T=t))}{\\B P(T=t)} \\leq\n  \\frac{\\B E h(\\b Z_I)}{\\B P(T=t)}.\n\\]\n\\hfill \\mbox{\\rule{0.5em}{0.5em}}\n\n\\subsection{Example: partitions of a set}\nRecall that partitions of a set is the assembly with $m_i = 1$ for all\n$i$.  \nFollowing the discussion in Section \\ref{sect5.3}\nwe take $x \\equiv x(n)\n=\\log n - \\log \\log n + \\cdots$ to be the solution on $x\ne^x=n$, so that for\n$i=1,2,\\ldots,n$, $Z_i$ is Poisson distributed, with mean and variance\n$\\lambda_i = x^i\/i!.$  \nWith this choice of $x$, we have\n\\[\n\\B E T_n = \\sum_1^n i \\lambda_i \\sim x e^x  = n\n\\]\nand\n\\begin{equation}\\label{Moser2}\n\\sigma_n^2 \\equiv {\\rm var}(T_n) = \\sum_1^n i^2 \\lambda_i \\sim n \\log n.\n\\end{equation}\nBy combining (\\ref{assprobT=t}) with the asymptotics for the Bell numbers\ngiven in Moser and Wyman (1955), and simplifying, we see that\n\\begin{equation} \\label{Moser}\n\\B P(T_n = n) \\sim \\frac{1}{\\sqrt{2 \\pi n \\log n}} \\sim \\frac{1}{\\sqrt{2\n\\pi } \\ \\sigma_n},\n\\end{equation}\nwhich is easy to remember, since it agrees with what one would guess\nfrom the local central limit heuristic.\n\nWrite $U_n =Z_1+Z_2+\\cdots+Z_n$, so that \nthe total number of blocks $K_n$ satisfies $K_n \\stackrel{\\mbox{\\small d}}{=} (U_n|T_n=n)$.\nHarper (1967) proved that $K_n$ is\nasymptotically normal with mean $n\/x$ and variance $n\/x^2$.  We observe\nthat this contrasts with the unconditional behavior: $U_n$ is\nasymptotically normal with mean $n\/x$, like $K_n$, and variance $n\/x$,\nunlike $K_n$.  Since $U_n$ is\nPoisson, it has equal mean and variance. Harper's result says that\nconditioning on $T_n=n$ reduces the variance of $U_n$ by a factor\nasymptotic to $\\log n$. \n\nNote that $Z_1$ is Poisson with parameter $x \\sim \\log n$, and hence\nthe distribution of $Z_1$ is asymptotically normal with mean and\nvariance $\\log n$.  Note also that \nthe Poisson parameters $\\lambda_i = x^i\/i!$ are themselves\nproportional to $\\B P(Z_1=i)$; in fact for $i \\geq 1$\n\\[\n\\lambda_i = e^x \\B P(Z_1 =i) = \\frac{n}{x} \\B P(Z_1=i).\n\\]\nWe can use the normal\napproximation for $Z_1$ to see that, for fixed $a<b$, as $n \\rightarrow\n\\infty$, \n\\[\n\\sum_{a \\sqrt{\\log n} < i-\\log n < b \\sqrt{\\log n}} \\lambda_i \\ \\sim \\  \n  \\frac{n}{\\log n} \\ \\int_a^b \\frac{1}{\\sqrt{2 \\pi}} e^{-u^2\/2} du.\n\\]  \nInformally, the relatively \nlarge values of $\\lambda_i$ occur for $i$ within a few\n$\\sqrt{\\log n}$ of $\\log n$.\n\n\\subsubsection{The size of a randomly selected block}\nA result similar to the following appears  as Corollary 3.3 in DeLaurentis and Pittel\n(1983).\nThe size $D \\equiv D_n$ of ``a randomly selected component'' of a\nrandom assembly on $n$ elements is defined by a two step procedure:\nfirst pick a random assembly, then pick one of its $K_n$ components,\neach with probability $1\/K_n$.  The same definition applies to the case\nof random multisets or selections of weight $n$.  \n\nGiven $1 \\leq b \\leq n$, consider the functional $h:\\B Z_+^n \\rightarrow\n[0,1]$ defined by\n\\[\nh(\\b a) = \\left( \\sum_{i \\leq b} a_i \\right)  \\ \/ \n \\left( \\sum_{i \\leq n} a_i \\right) ,\n\\]\nwith $h(0,0,\\ldots,0)$ defined to be 1. \nThe distribution of the size of a randomly selected component is\ndetermined by\n\\[\n\\B P(D_n \\leq b) \\ = \\ \\B E h(\\b C(n)).\n\\]\nDefine $U_b = Z_1 + \\cdots + Z_b$, so that $h(Z_1,\\ldots, Z_n) =\nU_b\/U_n$ and \n\\[\n\\B P(D_n \\leq b) \\ = \\ \\B E h((Z_1,\\ldots,Z_n)|T_n=n) = \\B E \\left(\n\\left. \\frac{U_b}{U_n} \\right| T_n=n \\right) .\n\\]\nLet $\\epsilon >0$ and $\\rho >1$ be given.  Let $1 \\leq b \\leq n$ such that\n\\begin{equation} \\label{qrelation}\nq \\equiv \\B P(Z_1 \\leq b) \\ \\in [2 \\epsilon,1-2 \\epsilon].\n\\end{equation}\nNow for all $n \\geq n(\\epsilon,\\rho)$ we have $\\B E U_b > \\epsilon \\ n\n\/\\log n$ and $\\B E U_b \/ \\B E U_n \\ \\in [q\/\\rho,q \\rho]$.\nLarge deviation theory says that for $\\rho >1$ there is a constant\n$c=c(\\rho) >0$ such that if $Y$ is Poisson with parameter $\\lambda$,\nthen $\\B P(Y\/\\lambda \\leq 1\/\\rho) \\leq \\exp(-\\lambda c)$ and $\\B\nP(Y\/\\lambda \\geq \\rho) \\leq \\exp(-\\lambda c)$. (In fact, the optimal $c$\nis given by $c(\\rho)= \\min(1+\\rho \\log \\rho - \\rho, 1- \\rho^{-1} \\log\n\\rho - \\rho^{-1})$, with the two terms in the minimum corresponding\nrespectively  to large\ndeviations above the mean and below the mean.) Putting\nthese together, using the large deviation bounds once with $U_b$ as $Y$\nand a second time with $U_n$ as $Y$, we have for $n \\geq n(\\epsilon,\\rho)$\n\\[\n\\B P(\\frac{U_b}{U_n} \\notin [q\/\\rho^3,q \\rho^3] \\ ) \\leq\n 2 \\exp( -c(\\rho) \\epsilon \\ n \/ \\log n).\n\\]\nSince the functional $h$ takes values in $[0,1]$, this proves, for $n\n\\geq n(\\epsilon,\\rho)$,\n\\begin{equation}  \\label{with a rate}\n\\left| \\B P(D_n \\leq b) - q \\right| \\  \\leq q(\\rho^3 -1) +\n 2 \\exp( -c(\\rho) \\epsilon \\ n \/ \\log n)\/\\B P(T_n=n).\n\\end{equation} \nIn terms of Lemma \\ref{dumbdumb}, the above argument involves the\nfunctional $h^*$ defined by $h^*(\\b a) = \\mbox{\\bf 1} (h(\\b a) \\notin [q\/\\rho^3, q\n\\rho^3])$. The inequality (\\ref{with a rate}) not only proves that $D_n$\nis asymptotically normal with mean and variance $\\log n$, but also provides\nan upper bound on the Prohorov distance between the distributions\nof $D_n$ and  $Z_1$.\n\n\\subsubsection{The size of the block containing a given element}\nIn the case of\nassemblies, it is possible that someone describing ``a randomly selected\ncomponent'' has in mind the component containing a randomly selected\nelement, where the element and the assembly are chosen independently. \nThis includes, for example, the case where the element is\ndeterministically chosen, say it is always 1.  Let  $D^*_n$ be the size\nof the component containing 1, in a random assembly on the set $[n]$. \n\nThe two notions of ``a\nrandomly selected component'' can be very far apart.  For example, with\nrandom permutations, $D^*_n$ is uniformly distributed over\n$\\{1,2,\\ldots,n\\}$, while the size $D_n$ of a randomly selected cycle is\nsuch that $\\log D_n \/ \\log n$ is approximately uniformly distributed\nover $[0,1]$.  For random partitions of a set, the argument below proves\nthat $D_n$ and $D_n^*$ are close in distribution, because both\ndistributions are close to Poisson with parameter $x$, where $xe^x=n$. \n\n\nGiven $1 \\leq b \\leq n$, consider the functional $g:\\B Z_+^n \\rightarrow\n[0,1]$ defined by\n\\[\ng(\\b a) = \\frac{1}{n} \\sum_{i \\leq b}i a_i  .\n\\]\nThe distribution of the size of the component containing a given element\nis\ndetermined by\n\\[\n\\B P(D^*_n \\leq b) \\ = \\ \\B E g(\\b C(n)).\n\\]\nDefine $R_b = Z_1 + 2 Z_2 + \\cdots +b Z_b$, so that $g(Z_1,\\ldots, Z_n) =\nR_b\/n$ and \n\\[\n\\B P(D^*_n \\leq b) \\ = \\ \\B E g((Z_1,\\ldots,Z_n)|T_n=n) =\n \\B E (R_b \/n| T_n=n ) .\n\\]\nWith  $\\epsilon,\\rho,b,n$ and $q$ as above in (\\ref{qrelation}), and\nwith the same $c(\\rho)$ as above but with a different\n$n(\\epsilon,\\rho)$, for all $n \\geq n(\\epsilon,\\rho)$ we have\n $\\B E U_b > \\epsilon \\ n\n\/\\log n$ and $\\B E R_b \/ n \\ \\in [q\/\\rho,q \\rho]$.\nLarge deviation theory says that, with  $\\lambda= \\B E U_b$\nas the mean of an unweighted sum of independent Poissons, the weighted\nsum $Y=R_b$ satisfies  $\\B P(Y\/\\B E Y \\leq 1\/\\rho)\n \\leq \\exp(-\\lambda c)$ and $\\B P(Y\/\\B E Y \\geq \\rho)\n \\leq \\exp(-\\lambda c)$. Putting\nthese together, we have for $n \\geq n(\\epsilon,\\rho)$\n\\[\n\\B P(\\frac{R_b}{n} \\notin [q\/\\rho^2,q \\rho^2] \\ ) \\leq\n 2 \\exp( -c(\\rho) \\epsilon \\ n \/ \\log n).\n\\]\nSince the functional $g$ takes values in $[0,1]$, this proves, for $n\n\\geq n(\\epsilon,\\rho)$,\n\\begin{equation}  \\label{biased with a rate}\n\\left| \\B P(D^*_n \\leq b) - q \\right| \\  \\leq q(\\rho^2 -1) +\n 2 \\exp( -c(\\rho) \\epsilon \\ n \/ \\log n)\/\\B P(T_n=n).\n\\end{equation} \n\n\\subsubsection{The number of distinct block sizes}\n\n Odlyzko and Richmond (1985) prove that the number $J_n$ of distinct\nblock sizes in a random partition of the set $[n]$ is asymptotic to $e\n\\log n$ in expectation and in probability.  A stronger result can easily\nbe proved by overwhelming the conditioning.\n\nInformally, our argument is that for $1 \\leq i \\leq (e-\\epsilon)  \\log\nn$, the Poisson parameter $\\lambda_i=x^i\/i!$ is large, so that $\\B P(Z_i=0)$ is\nvery small, in fact small enough to overwhelm the conditioning on\n$\\{T_n=n \\}$, so that $\\B P(C_i(n)=0)$ is also very small, and we\ncan conclude $\\B P(C_i(n) = 0 $ for any $i \\leq (e-\\epsilon)  \\log n))\n\\rightarrow 0$.  This accounts for at least $(e-\\epsilon)  \\log n$\ndistinct block sizes.  On the other side, $\\sum_{i \\geq (e+\\epsilon) \n\\log n} \\B EZ_i$ is  small, hence\n for some $k=k(\\epsilon)$, $\\B P(Z_i >0$ for at least $k$ values\nof $i \\geq (e+\\epsilon)  \\log n)$ is very small, \nin fact small enough to overwhelm\nthe conditioning (using roughly $k=1\/(2\\epsilon)$.)\nWe conclude $\\B P(C_i(n) > 0$ for at least $k$\n values $i \\geq (e+\\epsilon) \\log n) \\rightarrow 0$. Our result, that for any\n$\\epsilon > 0, \\B P(C_i(n)=0$ for any $i \\leq (e-\\epsilon)  \\log n$, or\n$C_i(n)>0$ for at least $k$ values \n $i \\geq (e+\\epsilon)  \\log n) \\rightarrow 0$,\nimplies but is not implied by the result that $J_n\/\\log n \\rightarrow e$ in\nprobability. Furthermore, the bounds supplied by Theorem \\ref{part\nparts} below imply that $J_n\/\\log n \\rightarrow e$ in $r$th mean for every $1\n\\leq r < \\infty$. The result that $\\B P(C_1(n)=0)\\rightarrow 0$ was\nproved in Sachkov (1974).\n\nIn a little more detail, observe that $\\B P(Z_1=0)=\\exp(-\\lambda_1)=\ne^{-x}=x\/n \\sim \\log n \/n$, which is smaller than the conditioning\nprobability, given by (\\ref{Moser}), by a factor on the order of\n$\\sqrt{n}\/(\\log n)^{3\/2}$.  The preceding argument is given in Sachkov\n(1974).  The Poisson parameters increase rapidly, so\n$\\B P(Z_2=0)=\\exp(-\\lambda_2)=\\exp(-x^2\/2)=(x\/n)^{x\/2}$, which decays\nfaster than any power of $n$.\n\nFor a more careful analysis of the boundary where the Poisson parameter\n$\\lambda_i$ changes from large to small, write $i=(x+d)e$, where\n$d=o(x)$. Recall $x \\sim \\log n$. \n Using Stirling's formula, and writing $\\approx$ for\n logarithmically asymptotic, we have $\\lambda_i = x^i\/i! \\sim\n(xe\/i)^i\/\\sqrt{2 \\pi i} = (x\/(x+d))^i \/ \\sqrt{2 \\pi i} \\approx $ $\n\\exp(-id\/x -\\log \\sqrt{i}) \\approx \n\\exp(- ed -\\frac{1}{2} \\log \\log n)$, so that the critical boundary for\n$i$,\ncorresponding to\n$d=-\\frac{1}{2e} \\log \\log n$, is at $c(n) \\equiv  xe - \\frac{1}{2} \\log\n\\log n$.  On the left side of this boundary \nthe argument via overwhelming the\nconditioning shows that $\\B P(C_i(n)=0$ for any $i<ex-\n(\\frac{3}{2}+\\epsilon) \\log \\log n) \\rightarrow 0$. The\nargument is very asymmetric between left and right: on the left, where\n$\\lambda_i$ is large, we use $\\B P(Z_i=0)=\\exp(-\\lambda_i)$, gaining the\nuse of an exponential; while on the right, where $\\lambda_i$ is small,\nwe use $\\B P(Z_i>0)<\\lambda_i$.  Thus in Theorem \\ref{part parts}, the\nleft boundary $a$ is an extra $(1+\\epsilon) \\log \\log n$ below $c(n)$,\nwhile the right boundary $b$ is an extra $\\epsilon \\log n$ above\n$c(n)$. \n\nThe results of the above discussion are summarized by the following \n\\begin{theorem}\\label{part parts}\nFor partitions of a set of size $n$, for $\\epsilon >0$, there are with\nhigh probability blocks of every size $i \\leq (e-\\epsilon)  \\log n$,\nand not many blocks of size $i \\geq (e+\\epsilon)  \\log n$.  More precisely,\nfor any $r<\\infty$ there exists $k=k(\\epsilon,r)<\\infty$ so that, as $n\n\\rightarrow \\infty$,\n\\[\n\\B P(C_1(n) = 0)  =  O((\\log n)^{3\/2}\/\\sqrt{n}),\n\\]\nwhile for $a \\equiv ex- (\\frac{3}{2}+\\epsilon) \\log \\log n)$\n\\[\n\\B P(C_i(n)=0 \\mbox{ for any } 2 \\leq i \\leq a) \\leq \\frac{1}{\\B P(T_n=n)} \\sum_2^a\ne^{-\\lambda_i} = o(n^{-r}),\n\\]\nand\n\\[\n\\B P(\\sum_{i \\geq b \\equiv (e+\\epsilon) \\log n} C_i(n) \\geq k)\n=O \\left(\\frac{1}{\\B P(T_n=n)}( \\sum_{i\\geq b} \\lambda_i)^k \\right)\n=o(n^{-r}),\n\\]\nwhere  $xe^x=n$, $\\lambda_i=x^i\/i!$,\nand $\\B P(T_n=n)$ satisfies (\\ref{Moser}).\n\\end{theorem}\n\\noindent{\\bf Proof\\ \\ }\nMost of the proof is contained in the informal discussion before the\ntheorem. For the second statement, it remains to check that $\\sum_2^a \\exp(-\n\\lambda_i)=o(n^{-r})$ for any $r$, which follows from an upper bound on\nthe first and last terms of the sum, which has at most $n$ terms, \ntogether with the observation that\nthe $\\lambda_2 < \\lambda_3 < \\cdots < \\lambda_{\\lfloor x \\rfloor} \\geq\n\\cdots > \\lambda_{\\lfloor a \\rfloor}$. For the third statement, we are\nmerely using the estimate, for $Y = \\sum_{i \\geq b} Z_i$, which is\nPoisson with small parameter $\\lambda$, that $\\B P(Y \\geq k)\n=O(\\lambda^k)$ as $\\lambda \\rightarrow 0$. Note that ${\\Bbb E} Y \\approx {\\Bbb E}\nZ_{\\lceil b \\rceil} \\equiv \\lambda_{\\lceil b \\rceil} \\approx (xe\/b)^b\n\\approx (1 + \\epsilon\/e)^{- b} < n^{-\\epsilon}$.\n\\hfill \\mbox{\\rule{0.5em}{0.5em}}\n\n\nThe above argument by overwhelming the conditioning is crude but easy to\nuse because it gives away a factor of $\\B P(T_n=n)$, when in fact the\nevent $\\{T_n=n \\}$ is approximately independent of the events involving\n$\\{Z_i>0\\}$ for large $i$. An effective way to quantify and handle this\napproximate independence is the total variation method outlined in\nsections 3 and 4. Sachkov (1974) analyzed the size $L_n$ of the \nlargest block of a random \npartition, and gave its approximate distribution.  Writing $L_n = h(\\b\nC(n))$ where $h(a_1,\\ldots,a_n) = \\max(i:a_i>0)$, Sachkov's result can\nbe paraphrased as $d_{TV}(L_n, h(\\b Z_n)) \\rightarrow 0$.  Note that the\nnumber $J_n$ of distinct block sizes satisfies $J_n\n\\leq L_n$ always. \nUsing $B=\\{ i\\leq n: i>ex - 2\\log \\log n \\}$ for example, it should be\npossible to prove that $d_{TV}(\\b C_B,\\b Z_B) \\rightarrow 0$.  Then, by\ncomparison of $J_n = h(\\b C(n))$ with $h(Z_1,\\ldots,Z_n) = \\sum \n\\mbox{\\bf 1}(Z_i>0)$, it would follow that, with centering constants $c(n)\n\\equiv ex -\\frac{1}{2e} \\log \\log n$, the family of random variable\n  $\\{ J_n - c(n) \\}$\nis tight, and the family  $\\{L_n-J_n\\}$\nis tight;  and for each family, \nalong a subsequence $n(k)$ there is convergence in\ndistribution if and only if the  $c(n(k)) \\mbox{ mod } 1$ converge.\n\n \n \n\n\n \\section{Dependent process approximations}\\label{sect12}\n\n\nFor the logarithmic class of structures discussed in Sections\n\\ref{logsect}, \\ref{esfsect}, and \\ref{sect5.2}, we have seen that the Ewens\nsampling formula (ESF) plays a crucial role. In the counting process for\nlarge components of\nlogarithmic combinatorial structures, there is substantial dependence;\nan appropriate comparison object is the dependent process of large\ncomponents in the ESF. For example, in Arratia, Barbour\nand Tavar\\'e (1993) it is shown that the process of counts of factors of large\ndegree in a random polynomial over a finite field is\nclose in total variation to the process of counts of large cycles in a\nrandom  permutation, corresponding to the ESF with parameter $\\theta =\n1$. In Arratia, Barbour and Tavar\\'e (1994), Stein's method is used to\nestablish an analogous result for all the logarithmic class, and\nsomewhat more generally. The basic  technique involving Stein's method\nspecialized to the compound Poisson is described in Barbour, Holst\nand Janson (1992, Chapter 10). \n\nOnce such bounds are available, it is  a simple matter to establish\napproximation results, with bounds, for other interesting functionals of\nthe large component counts of the combinatorial process. For example, \nthe Poisson-Dirichlet and GEM limits for random polynomials are established with\nmetric bounds in Arratia, Barbour and Tavar\\'e (1993). \nPoisson-Dirichlet limits for the logarithmic class are also discussed\nby Hansen (1993). \n\n\n\n\n\\section{References}\n\n\\begin{verse}\n\n\\bigskip\nAPOSTOL, T.M. (1976) {\\em An Introduction to Analytic Number Theory}, Springer\nVerlag, New York.\n \nARRATIA, R. and TAVAR\\'E, S. (1992a) The cycle structure of random\npermutations. {\\em Ann. Prob.} 20, 1567-1591.\n\nARRATIA, R. and TAVAR\\'E, S. 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{"text":"\\section{Introduction} \n\\label{sec:intro}\nMagnetic field is a key parameter in theories of stellar formation and\nevolution. In cool stars, it powers several activity phenomena, observed\non a wide range of wavelengths and timescales, which provide a rough proxy\nof the averaged field strength. The well-established rotation-activity\nrelation \\cite[e.g., ][]{Noyes84} supports the idea of a\ndynamo-generated magnetic field in these stars.\n\n\\cite{Larmor19} first proposed that the solar magnetic field could be\ninduced by plasma motions, already pointing out the importance of shear\n($\\Omega$ effect) to generate strong toroidal fields and thus explain\nHale's polarity law of sunspots. \\cite{Parker55} completed the basic\npicture of the solar dynamo by introducing the $\\alpha$ effect (convection\nmade cyclonic by the Coriolis force) to address the issues of Cowling's\nanti-dynamo theorem \\citep{Cowling34} and the regeneration of a poloidal\nfield component from a toroidal one. The $\\alpha\\Omega$ dynamo has\nthereafter been thoroughly debated and improved \\citep[e.g. ][]{Babcock61,\nLeighton69}. More recently, helioseismology has been able to probe the\nsolar interior and revealed the existence of a thin layer of strong shear\nlocated at the base of the convection zone: the tachocline\n\\cite[e.g.,][]{Spiegel92}. Although many aspects of the solar magnetism are\nstill not thoroughly understood, recent theoretical and numerical studies\nhave pointed out the crucial role of the tachocline as the place of\nstorage and amplification of strong toroidal fields \\cite[e.g.,\n][]{Ossendrijver03, Charbonneau05}.\n\nNew insight on dynamo processes can be gained from the exploration of magnetic\nfields of cool stars, probing the dynamo response in very non-solar regimes of\nparameters (e.g., fast rotation, deeper or shallower convection zone). M dwarfs\nare particularly interesting since those below $\\sim 0.35~\\hbox{${\\rm M}_{\\odot}$}$\n\\cite[e.g.,][]{Chabrier97} are fully convective and therefore do not possess a\ntachocline and presumably cannot host a solar-type dynamo. Yet, many M\ndwarfs are known to be active, and these stars follow the usual\nrotation-activity relation \\cite[e.g., ][]{Delfosse98, Reiners07, Kiraga07,\nWest04}. However the strong correlation between X-ray and radio luminosities\nestablished by \\cite{Guedel93} for stars of spectral types ranging form F to mid\nM, is no longer valid for very low mass dwarfs which exhibit very\nstrong radio emission whereas X-ray emissions dramatically drop\n\\cite[][]{Berger06}. Magnetic fields were also directly detected at photospheric\nlevel through Zeeman effect both in unpolarised \\cite[e.g., ][]{Saar85, Johns96,\nReiners06} and circularly polarised \\citep{Donati06} line profiles.\n\nSpectropolarimetry combined with tomographic imaging techniques is the optimal\ntechnique to investigate the magnetic topologies of M dwarfs (see\nSec.~\\ref{sec:model} for more details). By recovering the large-scale component\nof stellar magnetic fields, we can provide dynamo theorists with observables\ndirectly comparable with their modeling (axisymmetry, relative importance of the\npoloidal and toroidal components, characteristic scales...). Previous results\nhave already produced strong constraints: \\cite{Donati06} and \\cite{Morin08a}\n(hereafter M08a) demonstrated that the fully convective fast rotator V374~Peg is\nable to trigger a strong large-scale axisymmetric poloidal field steady on a\ntimescale of 1~year; and exhibits a very low level of differential rotation\n($\\sim \\frac{\\hbox{$d\\Omega_{\\odot}$}}{10}$).  From the analysis of a sample of early and mid M\ndwarfs  \\cite{Donati08b} and \\cite{Morin08b} (hereafter D08 and M08b\nrespectively) observed a strong change near the theoretical full-convection\nthreshold: while partly convective stars possess a weak non-axisymmetric field\nwith a significant toroidal component, fully convective ones exhibit strong\npoloidal axisymmetric dipole-like topologies. Differential rotation also drops\nby an order of magnitude across the boundary, the observed fully convective\nstars exhibit nearly solid body rotation.\nD08 and M08b also\nreport a sharp transition in the rotation-large-scale magnetic field \nrelation close to the full-convection boundary, whereas no such gap is visible\nin the rotation-X-ray relation. Considering that X-ray emission is a\ngood proxy for the total magnetic energy, it suggests that a sharp transition in\nthe characteristic scales of the magnetic field occurs near the full-convection\nlimit. This point was further confirmed for a few stars by \\cite{Reiners09b} who\nreport that the ratio of the magnetic fluxes measured from circularly-polarised\nand unpolarised lines dramatically changes across the fully convective limit.\nSeveral theoretical studies have addressed the challenging issue of dynamo\naction in fully convective stellar interiors. \\cite{Durney93} first proposed\nthat without a tachocline of shear, convection and turbulence should play the\nmain role at the expanse of differential rotation, generating a small-scale\nfield. \\cite{Kuker99} and \\cite{Chabrier06} performed mean-field modeling of\ndynamo action in fully convective stars and found purely non-axisymmetric\n$\\alpha^2$ solutions, indicating that these objects can sustain large-scale\nmagnetic fields. Subsequent direct numerical simulations by \\cite{Dobler06} and\n\\cite{Browning08} both realized large-scale dynamo action with a significant\naxisymmetric component of the resulting magnetic field. The latter also achieved\nmagnetic energy in equipartition with kinetic energy and therefore Maxwell\nstresses strong enough to quench differential rotation, resulting in nearly\nsolid-body rotation. Despite these recent advances, the precise causes of\ndifferences between dynamo in fully and partly convective stars are not\ncompletely understood, and theoretical studies need observational guidance.\n\nIn the present paper, we extend our spectropolarimetric study to 11 late M\ndwarfs (spectral types ranging from M5 to M8). After a brief presentation of the\nstellar sample and of spectropolarimetric observations, we describe the main\nprinciples of the tomographic imaging process. The Zeeman-Doppler Imaging (ZDI\nhereafter) analysis is then detailed for 6 stars. For 3 other late M\ndwarfs, it is not possible to derive a definitive magnetic map because of the\nvery low level of variability in the Stokes~$V$ signatures, but we can still\ninfer some information about their magnetic topologies. For the very faint\nstars  VB~8 and VB~10, the noise level is too high to allow definite detection\nof the circularly polarised signatures in individual LSD spectra. By averaging\nthe spectra of each data set, we marginally detect a large-scale magnetic field\non VB~10, and show a tentative ZDI reconstruction. We finally discuss these\nresults and conclude on the implications of our study for the understanding of\ndynamo processes in fully convective stars.\n\n\\section{Observations}\n\\subsection{Presentation of the sample}\n\\label{sec:obs-sample}\nFor this first spectropolarimetric survey, we selected 23 active main-sequence\nM dwarfs, mostly from the rotation-activity study by \\cite{Delfosse98},\ncovering a wide range of masses and rotation periods (although for a given mass\nthe extent in rotation period is rather restricted). In the present paper we\nfocus on the low-mass end of the sample (0.08--0.20~\\hbox{${\\rm M}_{\\odot}$}): GJ~51, WX~UMa,\nDX~Cnc, GJ~1245~B, GJ~1156 and GJ~3622 are thoroughly studied with tomographic\nimaging techniques. We also present a brief study of the large-scale magnetic\ntopologies of GJ~1224, GJ~1154~A, CN~Leo, VB~8 and VB~10\n(section~\\ref{sec:other}).\n\nAll these stars are known to show signs of activity in H$\\alpha$ or X-rays\n\\cite[e.g.][]{Gizis02, Schmitt04}, and photospheric magnetic fields have been\nmeasured on most of them from the analysis of Zeeman broadening in molecular\nbands (see below). The main properties of the sample,\ninferred from this work or collected from previous ones, are shown in\nTab.~\\ref{tab:sample}.\n\n\\begin{table*}\n\\begin{center}\n\\caption[]{Fundamental parameters of the stellar sample. Spectral types\n are taken from \\cite{Reid95}. Formal error bars derived from our study\nare mentioned between brackets for \\hbox{$M_{\\star}$}, \\hbox{$R\\sin i$}, \\hbox{$R_{\\star}$}\\ (1$\\sigma$) and\n\\hbox{$P_{\\rm rot}$}\\ (3$\\sigma$), they apply to the last digit of the preceding number.\nSee Sec.~\\ref{sec:obs-sample}\nfor more details and a discussion about uncertainties.}\n\\begin{tabular}{cccccccccccc}\n \\hline\n Name & ST & \\hbox{$M_{\\star}$} & \\hbox{$v\\sin i$} & $Bf$ &  $\\hbox{$P_{\\rm rot}$}$  & $\\tau_c$ & $Ro$ & \\hbox{$R\\sin i$} &\n\\hbox{$R_{\\star}$} & $i$ \\\\\n & & (\\hbox{${\\rm M}_{\\odot}$}) & (\\hbox{km\\,s$^{-1}$}) & (\\hbox{$\\rm kG$}) & (d) & (d) &\n ($10^{-2}$) & (\\hbox{${\\rm R}_{\\odot}$}) & (\\hbox{${\\rm R}_{\\odot}$}) & (\\hbox{$^\\circ$}) \\\\ \n \\hline\n GJ~51 & M5 & 0.21 (3) & 12 & -- & 1.02 (1) & 83 & 1.2 & 0.24 (2) & 0.22 (3) &\n60 \\\\\nGJ~1156 & M5 & 0.14 (1) & 17$^b$ & 2.1$^b$ & 0.491 (2) & 94 & 0.5 & 0.16 \n($<1$)& 0.16 (1) & 60 \\\\\nGJ~1245~B & M5.5 & 0.12 ($<1$) & 7$^a$ & 1.7$^a$ & 0.71 (1) & 97 & 0.7 & 0.10\n(2) & 0.14 ($<1$) & 40 \\\\\nWX~UMa & M6 & 0.10 ($<1$) & $5^b$ & $> 3.9^b$ & 0.78 (2) & 100 & 0.8 & $0.07$\n(1)\n& 0.12 ($<1$) & 40\\\\\nDX~Cnc & M6 & 0.10 ($1$) & $13^a$  & 1.7$^a$ & 0.46 (1) & 100 & 0.5 & $0.07$ (1)\n& 0.11 ($<1$) &\n60\\\\\nGJ~3622 & M6.5 & 0.09 ($<1$) & $3^c$ & -- & 1.5(2) & 101 & 1.5 & 0.09 (6) & 0.11\n($<1$) & 60 \\\\\n\\hline\nGJ~1154~A & M5 & 0.18 (1) & 6$^b$ & 2.1$^b$ & $\\leq1.7 $&88 &$\\leq1.9$\n&--& 0.20 (1)  &--\\\\\nGJ~1224 & M4.5 & 0.15 (1) & $\\leq 3^a$ & 2.7$^a$ & $\\leq4.3$ & 93 &$\\leq4.6$\n&--& 0.17 ($1$) &--\\\\\nCN~Leo & M5.5 & 0.10 ($<1$) & 3$^a$ & 2.4$^a$ & $\\leq2.0$ & 99 &$\\leq2.0$\n&--& 0.12 ($<1$) &--\\\\\nVB~8\u00a0& M7 & 0.09 ($<1$) & 5$^a$ & 2.3$^a$ & $\\leq1.0$ & 101 &$\\leq1.0$ &--& \n0.10 ($<1$) &--\\\\\nVB~10 & M8 & 0.08 ($<1$) & 6$^a$ & 1.3$^a$ & $\\leq0.8$ & 102 &$\\leq0.8$\n&--& 0.09 ($<1$) &--\\\\\n\\hline\n\\label{tab:sample}\n\\end{tabular}\n\\end{center}\n{\\flushleft\n$^a$ \\cite{Reiners07}\\\\\n$^b$ \\cite{Reiners09}\\\\\n$^c$ \\cite{Mohanty03}\\\\\n}\n\\end{table*}\n\nStellar masses are computed from the mass-luminosity relation derived by\n\\cite{Delfosse00}, based on $J$ band absolute magnitude inferred from apparent\nmagnitude measurements of 2MASS \\citep[][]{Cutri03} and \\emph{Hipparcos}\nparallaxes \\citep[][]{ESA97}. Formal error bars, as derived from\nuncertainties on these measurements are mentioned. The\nintrinsic dispersion of the relation is estimated to be lower than 10~$\\%$.\nRadius and bolometric luminosity suited to the stellar mass are computed from\nNextGen models \\citep[][]{Baraffe98}, formal error-bars on stellar mass\nare propagated. The accuracy of these models for active M dwarfs is a debated\nsubject \\cite[\\eg][]{Ribas06}, but recent studies indicate that the agreement\nwith observations is very good for late M dwarfs \\cite[][]{Demory09}.\nFor all stars except GJ~51, projected rotational velocities (\\hbox{$v\\sin i$})\nare available from previous spectroscopic studies. The uncertainties on \\hbox{$v\\sin i$}\\\nare typically equal to 1~\\hbox{km\\,s$^{-1}$}. For each star, we also mention the\nrotation period (\\hbox{$P_{\\rm rot}$}) derived from our analysis (see\nSec.~\\ref{sec:techniques-period}), and \\hbox{$R\\sin i$}\\ is straightforwardly deduced\n(with propagated error bar). An estimate of the inclination angle of the\nrotation axis on the line-of-sight ($i$) is obtained by comparing \\hbox{$R\\sin i$}\\ and\nthe theoretical radius. With this estimate the typical error is of the order of\n10\\hbox{$^\\circ$}\\ for low and moderate inclinations, and 20\\hbox{$^\\circ$}\\ for high\ninclination angles, this is precise enough for the imaging process. The effect\nof these uncertainties on the reconstructed magnetic maps is discussed in\nSec.~\\ref{sec:techniques-uncert}.\n\nWe also mention unsigned magnetic fluxes from the literature (whenever\navailable in \\citealt{Reiners07} and \\citealt{Reiners09}) empirically derived\nfrom unpolarised molecular (FeH) line profiles. These estimates result from the\ncomparison with reference spectra of an active and an inactive star (corrected\nfor spectral type and rotational broadening) which are used to calibrate the\nrelation between broadening of magnetically sensitive lines and magnetic flux\n\\cite[][]{Reiners06}. The authors estimate that the precision of the\nmethod lies in the 0.5--1~\\hbox{$\\rm kG$}\\ range. As this method is not sensitive to the\nvector properties of the magnetic field, the $Bf$ values reflect the overall\nmagnetic flux on the surface of the star. Since Stokes~$V$ signatures\ncorresponding to neighbouring zones of fields with opposite polarities cancel\neach other, our spectropolarimetric measurements are not sensitive to tangled\nfields and only recover the uncancelled magnetic flux corresponding to the\nlarge-scale component of the magnetic topology. Therefore the ratio of both\nmagnetic fluxes is a clear indication of the degree of organization of the\nobserved magnetic field.\n\nThe convective Rossby number ($Ro$), which is the ratio of the rotation\nperiod and the convective turnover time, is believed to be the relevant\nparameter to study the impact of rotation on dynamo action in cool stars\n\\cite[][]{Noyes84}. In Table~\\ref{tab:sample}, we mention Rossby numbers based\non empirical convective turnover times derived by \\cite{Kiraga07} from the\nrotation--activity relation in X-rays. These turnover times are \\textit{ad-hoc}\nfitting parameters that reflect more the relation between activity and rotation\nat a given mass than an actual turnover time at a specific depth in the\nconvection zone. However, the resulting Rossby numbers allow us to compare the\neffect of rotation on magnetic field generation in stars having different\nmasses.\n\n\\subsection{Instrumental setup and data reduction}\n\\label{sec:obs-red}\nObservations presented here were collected between June 2006\nand July 2009 with the ESPaDOnS spectropolarimeter at CFHT . ESPaDOnS provides\nfull coverage of the optical domain (370 to 1,\\,000~nm) in a single exposure, at\na resolving power of 65,\\,000, with a peak efficiency of 15\\% (telescope and\ndetector included). \n\nData Reduction is carried out with \\textsc{libre-esprit}, a fully-automated\ndedicated pipeline provided to ESPaDOnS and NARVAL users, that performs optimal\nextraction of the spectra following the procedure described in \\cite{Donati97}\nthat is based on \\cite{Horne86} and \\cite{Marsh89}. Each set of 4 individual\nsub-exposures taken in different polarimetric configuration are combined\ntogether to produce Stokes~$I$ (unpolarised intensity) and $V$ (circularly\npolarised) spectra, so that all spurious polarisation signatures are cancelled\nto first order \\citep{Semel93, Donati97}.  In addition, all spectra are\nautomatically corrected for spectral shifts resulting from instrumental effects\n(e.g., mechanical flexures, temperature or pressure variations) using telluric\nlines as a reference. Though not perfect, this procedure allows spectra to be\nsecured with a radial velocity (RV) internal precision of better than\n$0.030~\\hbox{km\\,s$^{-1}$}$ \\citep[e.g.,][]{Moutou07}.\n\nThe peak signal-to-noise ratios (\\hbox{S\/N} ) per 2.6~\\hbox{km\\,s$^{-1}$}\\ velocity bin range from 51\nto 245, mostly depending on the magnitude of the target and the weather\nconditions. An overview of the observations is presented in Table~\\ref{tab:obs},\nthe full journal of observations is available in the electronic\nversion.\nUsing the Least-squares deconvolution technique\n\\cite[LSD, ][]{Donati97}, polarimetric information is extracted from most\nphotospheric atomic lines and gathered into a single synthetic profile of\ncentral wavelength $\\lambda_0 = 750~{\\rm nm}$ (800~nm for VB~8 and 10). The\ncorresponding effective Land\\'e factor $g_{\\rm eff}$ (computed as a weighted\naverage on available lines) is close to 1.2 for all the stars of our\nsample. The line list for LSD was computed from an Atlas9 local thermodynamic\nequilibrium model \\citep{Kurucz93}\nmatching the properties of our whole sample, and contains about\n$5,\\,000$ moderate to strong atomic lines. We notice a multiplex gain of about\n10 (5 for VB~8 and 10) with respect to the peak \\hbox{S\/N}\\ of the\nindividual spectra of our sample. Although all the stars in the sample are\nactive, some exhibit Stokes~$V$ LSD signatures just above noise level (e.g. \nDX~Cnc), whereas on others we detect very strong signatures, with peak-to-peak\namplitudes as high as 1.8\\% of the unpolarised continuum level (for WX~UMa). \nTemporal variations, due to rotational modulation, of the Zeeman signatures is\nobvious for some stars, whereas it is very weak on others, depending \\eg on the\ninclination angle of their rotation axis with respect to the line of sight,\nthe complexity and the degree of axisymmetry of their magnetic topology.\n\nFor each observation we compute the corresponding longitudinal magnetic\nfield (i.e. the line of sight projection) from the Stokes $I$ and $V$\nLSD profiles through the relation:\n\n\\begin{equation}\n B_l({\\rm G}) = -2.14 \\times 10^{11} \\frac{\\displaystyle\\int v\\,V(v)\n\\,{\\rm d}v}{\\lambda_0\\,g_{\\rm eff}\\,c \\displaystyle\\int\n\\left[I_c-I(v)\\right] {\\rm d}v } \\, ,\n\\label{eq:bl}\n\\end{equation}\n\\citep[][]{Rees79, Donati97, Wade00} where $v$ is the radial velocity in\nthe star's rest frame, $\\lambda_0$, in nm, is the mean wavelength of the\nLSD profile, $c$ is the velocity of light in vacuum in the same unit as\n$v$, $g_{\\rm eff}$ is the value of the mean Land\\'e factor of the\nLSD line, and $I_c$ the continuum level.\n\nIn the rest of the paper, all data are phased according to the following\nephemeris:\n\\begin{equation}\n{\\rm HJD} = {\\rm HJD}_0 + \\hbox{$P_{\\rm rot}$} E,\n\\label{eq:eph}\n\\end{equation}\nwhere ${\\rm HJD}_0=2\\,453\\,850$ for WX~UMa, and ${\\rm HJD}_0=2\\,453\\,950$ for\nthe other stars; and $\\hbox{$P_{\\rm rot}$}$ is the rotational period used as an input for ZDI\nand given in Table~\\ref{tab:sample}.\n\n\n\\begin{table*}\n\\begin{center}\n\\caption[]{Synthetic journal of observations. Observation year and\nnumber of spectra collected are given in columns 2 and 3. Columns 4\nand 5 respectively list the peak signal to noise ratio (per 2.6~\\hbox{km\\,s$^{-1}$}\\\nvelocity bin) and the rms noise level (relative to the unpolarised\ncontinuum level and per 1.8~\\hbox{km\\,s$^{-1}$}\\ velocity bin) in the average circular\npolarisation profile produced by Least-Squares Deconvolution (see\ntext) --- we precise minimum and maximum values obtained\nfor each observing run. The average value and standard deviation of\nthe longitudinal magnetic field (see Eq.~\\ref{eq:bl}) and the radial velocity\nmeasurements are given in columns 6 and 7. The rotation cycle bounds\nof column 8 are computed with the rotation periods mentioned in\nTable~\\ref{tab:sample}. Complete observation logs are available in the\nelectronic version of the article.}\n\\begin{tabular}{cccccccc}\n\\hline\nName & Year & $n_{obs}$ & \\hbox{S\/N} & $\\sigma_{\\rm LSD}$ & $B_{\\ell}$ & RV &\nCycle \\\\\n &  & & & (\\hbox{$10^{-4} I_{\\rm c}$}) & (G) & (\\hbox{km\\,s$^{-1}$}) & \\\\\n\\hline \nGJ~51 & 2006 & 6 & 128--165 & 7.7--10.1 & -990 (313) & -5.52 (0.20) &\n5.0--9.9\\\\\n-- & 2007 & 9 & 159--198 & 5.7--7.0 & -1657 (280) & -6.36 (0.74) &\n412.7--418.7\\\\\n-- & 2008 & 9 & 118--181 & 6.9--10.3 & -1219 (407) & -6.60 (0.57) &\n788.0--794.1\\\\\nGJ~1156 & 2007 & 6 & 120--181 & 6.8--11.0 & 82 (72) & 5.96 (0.24) &\n431.9--442.0\\\\\n-- & 2008 & 5 & 127--158 & 10.4--8.4 & -47 (166) & 5.81 (0.13) &\n1091.6--1095.9\\\\\n-- & 2009 & 9 & 183--195 & 6.5--6.9 & 24 (111) & 5.73 (0.24) &\n1812.7--1817.0\\\\\nGJ~1245~B & 2006 & 6 & 158--191 & 7.1--8.8 & -52 (163) & 5.42(0.11) &\n4.2--14.0\\\\\n-- & 2007 & 6 & 182--226 & 5.7--7.4 & -17 (128) & 5.38 (0.10) &\n597.0--601.2\\\\\n-- & 2008 & 10 & 138--194 & 7.2--10.1 & -6 (67) & 5.46 (0.09) &\n1054.6--1057.8\\\\\nWX~UMa & 2006 & 8 & 67--142 & 19.8--9.6 & -1506 (453) & 70.25 (0.53)\n& 0.5--4.4\\\\\n-- & 2007 & 6 & 115--154 & 8.4--11.8 & -1757 (405) & 69.95 (0.06) &\n341.8--349.5\\\\\n-- & 2008 & 4 & 63--129 & 10.4--21.4 & -1811 (271) & 70.15 (0.24) &\n755.8--758.6\\\\\n-- & 2009 & 11 & 113--163 & 8.0--12.5 & -1496 (271) & 69.83 (0.21) &\n1214.9--1218.9\\\\\nDX~Cnc & 2007 & 5 & 119--179 & 8.3--12.3 & 132 (76) & 10.55 (0.07) &\n460.7--471.5\\\\\n-- & 2008 & 7 & 90--161 & 9.4--16.7 & 92 (52) & 10.44 (0.56) &\n1160.6--1169.5\\\\\n-- & 2009 & 9 & 106--187 & 7.8--14.8 & 67 (44) & 10.67 (0.10) &\n2012.8--2019.6\\\\\nGJ~3622 & 2008 & 8 & 128--167 & 8.9--11.1 & -32 (29) & 2.27 (0.28) &\n397.2--402.6 \\\\\n-- & 2009 & 6 & 101--162 & 9.3--15.0 & -26 (26) & 2.37 (0.05) &\n617.3--620.0\\\\ \n\\hline\nGJ~1154~A & 2007 & 6 & 118--167 & 7.0--10.5 & -714 (76) & -12.83 (0.11) & -- \\\\\n          & 2008 & 4 & 86--154 & 8.3--15.9 & -700 (75) & -12.92 (0.18) & -- \\\\\nGJ~1224 & 2008 & 12 & 51--185 & 5.8--19.7 & -563 (41) &  -32.68 (0.04) & -- \\\\\nCN~Leo & 2008 & 4 & 172--245 & 4.5--6.7 & -691 (54) & 19.62 (0.05) & -- \\\\\nVB~8 & 2009 & 9 & 83--107 & 15.7--20.0 & 29 (53) & 15.39 (0.11) & -- \\\\\nVB~10 & 2009 & 9 & 68--80 & 22.5--25.7 & 58 (61) & 36.23 (0.14) & -- \\\\\n\\hline \n\\label{tab:obs}\n\\end{tabular}\n\\end{center}\n\\end{table*}\n\n\\section{Data Modelling}\n\\label{sec:model}\nZeeman Doppler imaging \\cite[][]{Donati97b} aims at assessing stellar magnetic\ntopologies (at photospheric level), from the analysis of time-series of high\nspectral resolution spectropolarimetric observations. In this part we briefly\nremind the reader with the main properties of this technique and details of our\nimplementation. A more complete description can be found in M08b and references\ntherein.\n\nZDI is an inverse problem, the associated direct problem consists in computing\nthe Stokes~$I$ and $V$ spectra for a given magnetic map.\nThe stellar surface being sampled on a grid of $\\sim1,000$ cells, the local\nStokes $I$ and $V$ profiles are computed from a model based on\nUnno-Rachkovsky's equations, for a given magnetic map. The magnetic field is\ndecomposed into its poloidal and toroidal components and described as a set of\nspherical harmonics-like coefficients, as implemented by \\cite{Donati06b}. We\nintroduce two filling factors that account for subgrid cancellation of\npolarised signatures corresponding to fields of opposite polarities, and allows\nus to accurately reproduce the LSD line profiles (M08b). The stellar\nspectrum is then obtained by a disk integration taking into account Doppler\nshift and limb-darkening.\n\n\\subsection{Rotational modulation of polarised lines}\nIn the presence of a magnetic field one can observe the Zeeman effect\non spectral lines: (i) unpolarised lines (Stokes $I$) are broadened with respect\nto the null field configuration and (ii) polarised signatures (Stokes $Q$, $U$\nand $V$) show up. Here we only study circular polarisation (Stokes $V$), which\nis sensitive to the strength and polarity of the line-of-sight\nprojection of the field. Because of the combination of stellar rotation and\nDoppler effect:\n\\begin{itemize}\n  \\item The contribution of a photospheric region to the stellar spectrum is\ncorrelated with its longitude (at first order). Thus, as a magnetic region\ncrosses the stellar disk under the effect of rotation, the corresponding\npolarised signal migrates from the blue to the red wing of the line.\n  \\item The amplitude of this migration depends on the spot's latitude (no\nmigration for a polar spot, maximum migration from -\\hbox{$v\\sin i$}\\ to +\\hbox{$v\\sin i$}\\ for an\nequatorial one).\n  \\item The evolution of the signature during this migration (as\nthe angle between the field vector and the line of sight changes with rotation)\nis characteristic of the field orientation (e.g. signature of constant polarity\nfor radial field, and polarity reversal for azimuthal field).\n\\end{itemize}\nTherefore, from a series of polarised spectra providing even and dense sampling\nof stellar rotation, it is possible to reconstruct a map for the photospheric\nmagnetic field.\n\n\\subsection{Magnetic field reconstruction: inverse problem}\n\\label{sec:techniques-inverse}\nStarting from a null-field configuration, the series of spectra computed from a\ntest field is iteratively compared to the observed one, until a given \\hbox{$\\chi^2_r$}\\\nlevel is reached. As the problem is partly ill-posed (several magnetic\nconfigurations can match a data set equally well), the maximum entropy solution\nis selected. The spatial resolution of ZDI depends on \\hbox{$v\\sin i$}\\ as a rule of\nthumb. The highest degree available in the field reconstruction is:\n\n\\begin{equation}\n  \\ell_{max} \\simeq \\max(\\frac{2\\pi\\hbox{$v\\sin i$}}{W}\\,;\\,\\ell_{\\min}) %\n\\label{eq:lmax}\n\\end{equation}\nwhere $W$ is the unpolarised local profile width ($\\sim 9~\\hbox{km\\,s$^{-1}$}$ for an inactive\nnon-rotating M dwarf, M08a). The first term in the max function\ncorresponds to the limit of high \\hbox{$v\\sin i$}, when line broadening is mostly due to\nrotation and the line profile can actually be seen as one 1-D map of the\nphotospheric magnetic field. The second term, $\\ell_{\\min}$ is the minimum\nresolution available in the low \\hbox{$v\\sin i$}\\ limit, when Doppler shift is small and\nthe information on the field topology mostly comes from the temporal evolution\nof the shape and amplitude of the polarised signatures. $\\ell_{\\min}$ can range\nfrom 4 to 8 mostly depending on the signal to noise ratio of the data (D08,\nM08b).\n\n\\subsection{Period determination}\n\\label{sec:techniques-period}\nSince ZDI is based on the analysis of rotational modulation, important inputs\nof the code are: the projected equatorial velocity (\\hbox{$v\\sin i$}), the inclination\nangle of the rotation axis with respect to the line of sight ($i$) and the\nrotation period (\\hbox{$P_{\\rm rot}$}). As no previous definite period measurement\nexists for any star of our late M subsample, we use ZDI to provide a constraint\non this parameter. For each star, given the \\hbox{$v\\sin i$}\\ and the theoretical radius\n(corresponding to the stellar mass) we can derive an estimate of the maximum\nvalue for the rotation period as : \n\\begin{equation}\n  P_{\\max} = 50.6145 \\times \\frac{\\hbox{$R_{\\star}$}}{\\hbox{$v\\sin i$}} ,\n\\end{equation}\nwhere $P_{\\max}$ is expressed in days, $\\hbox{$R_{\\star}$}$ in unit of \\hbox{${\\rm R}_{\\odot}$}\\ and \\hbox{$v\\sin i$}\\\nin \\hbox{km\\,s$^{-1}$}. We test period values in a reasonable range ($<1.2\\times P_{\\max}$),\nand derive the most probable period as the one resulting in the minimum \\hbox{$\\chi^2_r$}\\\nat a given informational content (i.e. a given averaged magnetic flux value).\nWe try to resolve aliases by comparing the multiple data sets available for each\nstar. \nBy fitting a parabola to the resulting \\hbox{$\\chi^2_r$}\\ curve close to the minimum, we\nderive the optimal value of \\hbox{$P_{\\rm rot}$}\\ and the associated formal error bar (see\nM08b for more details). Since several data sets are available for each star, we\nmention the smallest 3$\\sigma$ error bar in Tab.~\\ref{tab:sample}. For all the\nstudied stars periods inferred from different data sets are compatible with each\nother within the width of the associated error bars.\n\n\\subsection{Uncertainties on the magnetic maps}\n\\label{sec:techniques-uncert}\nDue to the use of a maximum entropy method, magnetic maps reconstructed with ZDI\nare optimal in the sense that any feature present in the map is actually\nrequired to fit the data. However this method does not allow us to derive\nformal error bars on the reconstructed maps.\n\nNumerical experiments demonstrate that ZDI provides reliable maps and\nis robust with respect to reasonable uncertainties on various parameters and \ndata incompleteness \\cite[e.g., ][]{Donati97b}. \nIn addition, our implementation based on spherical harmonics and\npoloidal\/toroidal decomposition successfully reconstructs global topologies such\nas low-degree multipoles, as well as more complex configurations \\cite[e.g.,\n][]{Donati06b}.\nWe also find that ZDI maps are robust with respect to the selected entropy\nweighting scheme, provided that phase coverage is complete enough.\n\nWe try to assess the effects of the uncertainties on the input parameters (see\nSec.~\\ref{sec:obs-sample}), on the derived magnetic quantities in a similar way\nas \\cite{Petit08}. For each data set we perform several reconstructions with\ninput parameters \\hbox{$v\\sin i$}, $i$, and \\hbox{$P_{\\rm rot}$}\\ varying over the width of the error bars\nand check the resulting map and its properties, thus providing a quantitative\nanalysis of the robustness of our reconstructions to these uncertainties. We\ntherefore obtain ``variability bars'' rather than formal error bars. Given\nthe small uncertainties on the rotation period, varying this parameter within\nthe 3$\\sigma$ error bar has negligible effect on the reconstructed maps. The\nmagnetic quantities listed in Tab.~\\ref{tab:syn} to characterize the repartition\nof magnetic energy into different components are affected in different\nways by variations of the input parameters. In particular, the decomposition\nbetween poloidal and toroidal energy is very robust, the observed variation is\nless than 10~\\% of the reconstructed magnetic energy. The fraction of\nmagnetic energy in axisymmetric modes varies by up to 20~\\% due to\nuncertainties on input parameters. The most important effect observed is a\ncross-talk between the dipole component and higher degree multipoles (in\nparticular the quadrupole), the variation of the fraction of energy in the\ndipole is in the 10--30\\% range. The variability bars on the reconstructed\nmagnetic flux are of the order of 30~\\%.\n\n\\section{GJ~51}\n\\label{sec:gj51}\n\\begin{figure*}\n\\begin{center}\n  \\includegraphics[height=0.40\\textheight]{fig_arxiv\/gl51_spec_060708.ps}\n\\end{center}\n \\caption[]{Time-series of Stokes $V$ profiles of GJ~51, in the\nrest-frame of the star, from our 2006 (left-hand column), 2007 (middle\ncolumn) and 2008 (right-hand column) data sets. Synthetic profiles\ncorresponding to our magnetic models (red lines) are superimposed to the\nobserved LSD profiles (black lines). Left to each profile a\n$\\pm1\\sigma$ error bar is shown. The rotational phase and cycle of each\nobservation is also mentioned right to each profile.\nSuccessive profiles are shifted vertically for clarity purposes and the\nassociated reference levels ($V=0$) are plotted as dotted lines.}\n\\label{fig:gj51_spec}\n\\end{figure*}\n\n\\begin{figure*}\n\\begin{center}\n\\includegraphics[height=0.40\\textheight]{fig_arxiv\/gl51_map_06nt.ps}\\hspace{\n0.5cm }\n\\includegraphics[height=0.40\\textheight]{fig_arxiv\/gl51_map_07nt.ps}\\hspace{\n0.5cm }\n  \\includegraphics[height=0.40\\textheight]{fig_arxiv\/gl51_map_08nt.ps}%\n  \\hspace{\\stretch{4}}\n\\end{center}\n\\caption[]{Surface magnetic flux of GJ~51 as derived from our 2006, (left-hand\ncolumn), 2007 (middle column) and 2008 (right-hand column) data sets.\nFor GJ~51, the imaging process is adapted to preferentially converge\ntoward a mostly axisymmetric solution in order to resolve the ambiguity due to\npoor phase coverage (see text). The three components of the field in spherical\ncoordinates are displayed from top to bottom (flux values labelled in G).  The\nstar is shown in flattened polar projection down to latitudes of $-30\\hbox{$^\\circ$}$,\nwith the equator depicted as a bold circle and parallels as dashed circles.\nRadial ticks around each plot indicate phases of observations.}\n\\label{fig:gj51_map}\n\\end{figure*}\n\nWe have acquired a total of 24 spectra on GJ~51 split in 3 series collected on 3\nsuccessive years (see Tab.~\\ref{tab:obs}). All\nStokes~$V$ spectra exhibit a strong signature of constant polarity (radial field\ndirected toward the star). Temporal variation inside each data set is\ndetectable and likely due to rotational modulation. It mainly consists of an\nevolution of the signature's amplitude. To our knowledge, no previously\npublished measurement of \\hbox{$P_{\\rm rot}$}\\ or \\hbox{$v\\sin i$}\\ exist for this star. From our LSD\nStokes $I$ profiles we measure mean RV values of $-5.52$, $-6.36$, and\n$-6.60~\\hbox{km\\,s$^{-1}$}$ in\n2006, 2007 and 2008, respectively. These values are in agreement with the\npreviously published $\\hbox{${\\rm RV}$}=-7.3~\\hbox{km\\,s$^{-1}$}$ \\cite[][]{Gizis02}. Our RV measurements\nalso reveal a drift between the 3 epochs, too large to be due to\nconvection and that may indicate a companion orbiting around this star. We also\nobserve strong RV temporal variations inside each data set presumably due to\nmagnetic activity (see Tab.~\\ref{tab:obs}),\nwell above the intrinsic precision of the instrument (see\nSec.~\\ref{sec:obs-red}). Our analysis of both the unpolarised and polarised\nspectra leads us to $\\hbox{$v\\sin i$}=12~\\hbox{km\\,s$^{-1}$}$ and $\\hbox{$P_{\\rm rot}$}=1.02~\\hbox{$\\rm d$}$, this is close to the\n1.06~d period derived from the MEarth\\footnote{\\cite{Irwin09}} photometric data\n(J.~Irwin, private communication). From these values we infer\n$\\hbox{$R\\sin i$}=0.24~\\hbox{${\\rm R}_{\\odot}$}$, whereas for $\\hbox{$M_{\\star}$}=0.20~\\hbox{${\\rm M}_{\\odot}$}$ evolutionary models\nexpect $\\hbox{$R_{\\star}$}=0.21~\\hbox{${\\rm R}_{\\odot}$}$. We conclude that the inclination angle of the\nrotation axis with respect to the line-of-sight must be high, and set\n$i=60\\hbox{$^\\circ$}$ for ZDI. We note that for $\\hbox{$P_{\\rm rot}$}=1.02~\\hbox{$\\rm d$}$ none of our data set\nprovide a complete sampling of the stellar rotation, the best one being\nthe 2008 data set which covers 30\\% of the rotation cycle. Despite this poor\nphase coverage, Stokes~$V$ profiles collected at the 3 epochs are similar,\nsuggesting that the magnetic field is stable and mostly axisymmetric.\n\n\\begin{table}\n\\begin{center}\n\\caption[]{Fit achieved for the 3 spectral time-series obtained on GJ~51. \nThe imaging process is guided towards a mostly axisymmetric solution, see text.\nIn columns 2 we give the maximum degree of spherical harmonics used for ZDI\nreconstruction. Columns 3--6 respectively list the initial \\hbox{$\\chi^2_r$}\\ (i.e.\nwithout magnetic field), the \\hbox{$\\chi^2_r$}\\ achieved with the imaging process, and the\naverage and peak value of the magnetic flux on the reconstructed map.}\n\\begin{tabular}{cccccc}\n\\hline\nEpoch & $\\ell_{ZDI}$ & ${\\hbox{$\\chi^2_r$}}_0$ & ${\\hbox{$\\chi^2_r$}}_f$ &\n$\\avg{B}$ & $B_{max}$ \\\\\n & & & & (\\hbox{$\\rm kG$}) & (\\hbox{$\\rm kG$}) \\\\\n\\hline\n2006 & 5 &  6.17 & 1.00 & 1.61 & 3.86 \\\\\n2007 & 5 & 24.53 & 1.00 & 1.58 & 5.02 \\\\\n2008 & 5 & 15.20 & 1.00 & 1.65 & 4.68 \\\\\n\\hline\n\\label{tab:gj51_fit}\n\\end{tabular}\n\\end{center}\n\\end{table}\n\nSetting $\\ell_{{\\rm ZDI}}=5$ for the magnetic field decomposition, it is\npossible to fit each of our 3 data sets with \\hbox{$\\chi^2_r$}=1. The resulting magnetic\nmaps feature a strong non-axisymmetric component with a dipole tilted toward the\nobserver, in particular those inferred from our 2006 and 2007 spectra. This is\nsurprising, it is indeed very unlikely that we observe three times GJ~51\nat the same phase (when the magnetic pole is crossing the line of\nsight).\n\nThe magnetic map reconstructed by ZDI is highly dependent on the precise\nform of the entropy, whereas it is generally not the case, indicating that\nthis reconstruction is particularly ill-posed. We suggest this is due to\nthe lack of information associated with the poor phase coverage, the\nreconstructed solution is strongly influenced by the maximum entropy constraint:\nthe resulting map is therefore mainly composed of a spot of radial field facing\nthe observer.\n\nWe perform another reconstruction of the magnetic field of GJ~51, with addition \nof \\emph{a priori} information in the process, so that it preferentially\nconverges toward a mostly axisymmetric solution, as far as it allows to fit the\ndata at the prescribed \\hbox{$\\chi^2$}\\ level. This is done by putting a strong\nentropy penalty on non-axisymmetric modes, similarly to the method used by\n\\cite{Donati08a} to drive the reconstructed topology towards antisymmetry with\nrespect to the center of the star. In these conditions, we can also fit our 3\ndata sets with \\hbox{$\\chi^2_r$}=1. The resulting magnetic maps for the 3 epochs are very\nsimilar (Fig.~\\ref{fig:gj51_map}). The corresponding synthetic spectra are\nplotted along with the data on Fig.~\\ref{fig:gj51_spec}. We find similar results\nat all epochs: the magnetic topology is almost purely poloidal and axisymmetric,\nit is mainly composed of a very strong dipole aligned with the rotation axis\n(see tables \\ref{tab:gj51_fit} and \\ref{tab:syn} for more details). The\nazimuthal and meridional component of the field somehow differ between the three\nepochs, but we consider that this is not significant given the weak constraint\nprovided by our data. These maps are not the only solution permitted by our\ndata, but we believe that they represent the most probable one. \n\nVariability bars are of the same order of magnitude as for the other stars. The\nalmost purely poloidal nature of the magnetic field is robust to uncertainties\non the input parameters $i$ and \\hbox{$v\\sin i$}. In addition, when varying these\nparameters, the topology always features a strong purely axisymmetric component\n(i.e. more than 45\\% of the magnetic energy is reconstructed in $m=0$ modes) and\nthe main reconstructed mode is the radial component of a dipole aligned with the\nrotation axis.\nHowever, the values mentioned in Tab.~\\ref{tab:syn} for GJ~51 should be\nconsidered cautiously as our data sets provide a weak constraint and the\nresulting magnetic maps are largely determined by the entropy function used.\nIn particular, the high values of longitudinal field measured (see\nTab.~\\ref{tab:obs}) indicate that the large-scale magnetic flux is indeed higher\nthan those of mid-M dwarfs studied in M08b. But it is not clear whether the\nlarge-magnetic flux of GJ~51 is actually larger than that of WX~UMa (see\nSec.~\\ref{sec:wxuma}).\nA definite confirmation\nof the magnetic topology of GJ~51 requires multi-site observations to obtain a\ncomplete sampling of the rotation cycle due to a period close to 1~\\hbox{$\\rm d$}. \n\n\\section{GJ~1156}\n\\label{sec:gj1156}\n\\begin{figure*}\n\\begin{center}\n  \\includegraphics[height=0.40\\textheight]{fig_arxiv\/gj1156_spec_070809e.ps}\n\\end{center}\n\\caption[]{Same as Fig.~\\ref{fig:gj51_spec} for GJ~1156 2007, 2008,\nand 2009 data sets (from left to right).}\n\\label{fig:gj1156_spec}\n\\end{figure*}\n\n\\begin{figure*}\n\\begin{center}\n\\includegraphics[height=0.40\\textheight]{fig_arxiv\/gj1156_map_07e.ps}\\hspace{\n0.5cm}\n\\includegraphics[height=0.40\\textheight]{fig_arxiv\/gj1156_map_08e.ps}\\hspace{\n0.5cm\n}\n\\includegraphics[height=0.40\\textheight]{fig_arxiv\/gj1156_map_09e.ps}%\n  \\hspace{\\stretch{4}}\n\\end{center}\n\\caption[]{Same as Fig.~\\ref{fig:gj51_map} for GJ~1156 2007, 2008, and\n2009 data sets (from left to right).}\n\\label{fig:gj1156_map}\n\\end{figure*}\n\nWe carried out 3 observing run on the flare star GJ~1156 --- in 2007, 2008 and\n2009 --- and obtained 20 pairs of Stokes $I$ and $V$ spectra.  The resulting LSD\npolarised signature is above noise level in nearly all observations. Temporal\nvariation is obvious on Figure~\\ref{fig:gj1156_spec}, we observe both variations\nof amplitude and polarity. We use $\\hbox{$v\\sin i$}=17~\\hbox{km\\,s$^{-1}$}$ \\cite[][]{Reiners07} which\nallows us to fit the observed polarised and unpolarised profiles, as opposed to\nthe previously reported value \\cite[$\\hbox{$v\\sin i$}=9.2~\\hbox{km\\,s$^{-1}$}$, ][]{Delfosse98}.\nWe derive $\\hbox{$P_{\\rm rot}$}=0.491~\\hbox{$\\rm d$}$, although 1\/3~\\hbox{$\\rm d$}\\ cannot be excluded as a\npossible period. This is confirmed by MEarth photometric periodogram which also\nexhibits a main peak at 0.491~\\hbox{$\\rm d$}\\ and another one at 1\/3~d (J.~Irwin, private\ncommunication). The corresponding $\\hbox{$R\\sin i$}$ is $0.16~\\hbox{${\\rm R}_{\\odot}$}$. As evolutionary\nmodels predict $\\hbox{$R_{\\star}$}=0.16~\\hbox{${\\rm R}_{\\odot}$}$, we set $i$=60\\hbox{$^\\circ$}\\ for ZDI.\nThe rotation period being close to a fraction of day our observations\n(especially those of 2007) do not provide an optimal sampling of rotation\nphases, only our 2008 data set provides a reasonable sampling on more than half\nof the rotation cycle.\n\n\\begin{table}\n\\begin{center}\n\\caption[]{Same as Tab.~\\ref{tab:gj51_fit} for GJ~1156.}\n\\begin{tabular}{cccccc}\n\\hline\nEpoch & $\\ell_{ZDI}$ & ${\\hbox{$\\chi^2_r$}}_0$ & ${\\hbox{$\\chi^2_r$}}_f$ &\n$\\avg{B}$ & $B_{max}$ \\\\\n & & & & (\\hbox{$\\rm kG$}) & (\\hbox{$\\rm kG$}) \\\\\n\\hline\n2007 & 6 & 1.95 & 0.95 & 0.06 & 0.32 \\\\\n2008 & 6 & 2.45 & 1.00 & 0.10 & 0.36 \\\\\n2009 & 6 & 2.07 & 1.00 & 0.09 & 0.36 \\\\\n\\hline\n\\label{tab:gj1156_fit}\n\\end{tabular}\n\\end{center}\n\\end{table}\n\nThe data sets can be fitted down to noise level with ZDI for the 3 epochs of\nobservation (see Tab.~\\ref{tab:gj1156_fit}).  The corresponding maps of surface\nmagnetic flux are displayed in Figure~\\ref{fig:gj1156_map}. The 3 maps exhibit\nsimilar properties: they mainly feature two radial field spots of opposite\npolarities. The magnetic topologies are thus predominantly poloidal (more than\n80~\\% of the overall magnetic energy in poloidal modes at all epochs) but\nfeature a significant toroidal component (in particular in 2007 and 2008), and\nnon-axisymmetric (more than 80~\\% of the overall magnetic energy in modes with\nazimuthal number $m > \\ell\/2$) as was expected from the polarity reversal of the\n$V$ signature inside each data set.  The lower average magnetic flux, as well as\nthe weakness of the spot of negative polarity (incoming field lines, in blue),\non the 2007 map can be attributed to poor phase coverage.\n\nThe reconstructed maps are quite robust to uncertainties on the input\nparameters (see Sec.~\\ref{sec:techniques-uncert}). In particular, when\nvarying these parameters, the fraction of magnetic energy reconstructed in \nnon-axisymmetric modes ($m>\\ell\/2$) is always higher than 70\\%.\nUsing the alternative values \\hbox{$P_{\\rm rot}$}=0.33~d and $i$=40\\hbox{$^\\circ$}, the reconstructed\nmagnetic maps do not change significantly. All the quantities listed in\ntable~\\ref{tab:syn} vary by less than 10~\\% of the total magnetic energy. The\nreconstructed magnetic flux variations range from 10 to 15~\\%. Considering these\nvalues therefore does not affect our conclusions.\n\n\\section{GJ~1245~B}\n\\begin{figure*}\n\\begin{center}\n  \\includegraphics[height=0.40\\textheight]{fig_arxiv\/gj1245b_spec_060708c.ps}\n\\end{center}\n\\caption[]{Same as Fig.~\\ref{fig:gj51_spec} for GJ~1245~B 2006, 2007, and\n2008 data sets (from left to right).}\n\\label{fig:gj1245b_spec}\n\\end{figure*}\n\n\\begin{figure*}\n\\begin{center}\n\\includegraphics[height=0.40\\textheight]{fig_arxiv\/gj1245b_map_06c_0.06.ps}\n\\hspace\n{ 0.5cm}\n\\includegraphics[height=0.40\\textheight]{fig_arxiv\/gj1245b_map_07c.ps}\\hspace{\n0.5cm }\n\\includegraphics[height=0.40\\textheight]{fig_arxiv\/gj1245b_map_08c.ps}%\n  \\hspace{\\stretch{4}}\n\\end{center}\n\\caption[]{Same as Fig.~\\ref{fig:gj51_map} for GJ~1245~B 2006, 2007, and\n2008 data sets (from left to right).}\n\\label{fig:gj1245b_map}\n\\end{figure*}\n\nThe M5.5 dwarf GJ~1245~B was observed during 3 successive years, for a\ntotal of 22 pairs of Stokes $I$ and $V$ spectra. The circularly polarised LSD\nsignatures (see Fig.~\\ref{fig:gj1245b_spec}) have a moderate amplitude and\nexhibit strong variability (amplitude, shape and polarity) during an observation\nrun --- presumably due to rotational modulation. Variability also seems\nimportant between the different epochs, in particular in the 2009 data set the\naverage amplitude of the signatures is significantly lower than at previous\nepochs, this is also visible in the longitudinal field measurements (see\nTab.~\\ref{tab:obs}).\n\nWe measure a mean $\\hbox{${\\rm RV}$} = 5.4~\\hbox{km\\,s$^{-1}$}$ with typical dispersions of $0.1~\\hbox{km\\,s$^{-1}$}$\n(see Tab.~\\ref{tab:obs}). This is compatible with the mean value of $5~\\hbox{km\\,s$^{-1}$}$\nreported by \\cite{Delfosse98}. We use $\\hbox{$v\\sin i$}=7~\\hbox{km\\,s$^{-1}$}$ (\\citealt{Delfosse98} ;\n\\citealt{Reiners07}); and find a rotation period of $0.71~\\hbox{$\\rm d$}$ corresponding to a\npeak in the photometric periodogram produced by the HATNet\n\\footnote{\\cite{Bakos04}} survey (J.~Hartman, private communication). With these\nvalues of \\hbox{$v\\sin i$}\\ and period, we find $\\hbox{$R\\sin i$}=0.10~\\hbox{${\\rm R}_{\\odot}$}$, as the NextGen\nevolutionary model predicts $\\hbox{$R_{\\star}$}=0.14~\\hbox{${\\rm R}_{\\odot}$}$ we set $i=40\\hbox{$^\\circ$}$ for our\nstudy.\n\n\\begin{table}\n\\begin{center}\n\\caption[]{Same as Tab.~\\ref{tab:gj51_fit} for GJ~1245~B.}\n\\begin{tabular}{cccccc}\n\\hline\nEpoch & $\\ell_{ZDI}$ & ${\\hbox{$\\chi^2_r$}}_0$ & ${\\hbox{$\\chi^2_r$}}_f$ &\n$\\avg{B}$ & $B_{max}$ \\\\\n & & & & (\\hbox{$\\rm kG$}) & (\\hbox{$\\rm kG$}) \\\\\n\\hline\n2006 & 4 & 4.41 & 1.00 & 0.17 & 0.47 \\\\\n2007 & 4 & 4.92 & 1.10 & 0.18 & 0.58 \\\\\n2008 & 4 & 1.81 & 1.00 & 0.06 & 0.22 \\\\\n\\hline\n\\label{tab:gj1245b_fit}\n\\end{tabular}\n\\end{center}\n\\end{table}\n\nRunning ZDI\\ on the LSD time-series, with the aforementioned parameters,\nwe can achieve a good fit for the 3 epochs (see Fig.~\\ref{fig:gj1245b_spec} and\nTab.~\\ref{tab:gj1245b_fit}). The reconstructed magnetic field\nsignificantly evolves between two successive epochs, in particular the\nreconstructed magnetic flux has strongly decreased between our 2008 and 2009\nobservations.\nHowever the magnetic topologies feature similar\nproperties: strong spots of radial field (although more than 40~\\% of the\nmagnetic energy lies in non-radial field structures); a mostly non-axisymmetric\nfield (more than 50\\% of the magnetic energy) and a significant toroidal\ncomponent (between 15 and 20\\% of the total magnetic energy).\nThe mainly non-axisymmetric nature of the magnetic field in 2006 and\n2008, as well as the presence of a significant toroidal component at all epochs\nare robust to uncertainties on stellar parameters (see\nSec.~\\ref{sec:techniques-uncert}).\n\n\\section{WX~UMA=GJ~412~B}\n\\label{sec:wxuma}\n\\begin{figure*}\n\\begin{center}\n  \\includegraphics[height=0.40\\textheight]{fig_arxiv\/gl412b_spec_06070809h.ps}\n\\end{center}\n\\caption[]{Same as Fig.~\\ref{fig:gj51_spec} for WX~UMa 2006, 2007, 2008\nand\n2009 data sets (from left to right).}\n\\label{fig:gj412b_spec}\n\\end{figure*}\n\n\\begin{figure*}\n\\begin{center}\n  \\includegraphics[height=0.40\\textheight]{fig_arxiv\/gl412b_map_06fb.ps}%\n  \\hspace{0.5cm }%\n  \\includegraphics[height=0.40\\textheight]{fig_arxiv\/gl412b_map_07h.ps}%\n  \\hspace{0.5cm} %\n  \\includegraphics[height=0.40\\textheight]{fig_arxiv\/gl412b_map_08h.ps}%\n  \\hspace{0.5cm}%\n  \\includegraphics[height=0.40\\textheight]{fig_arxiv\/gl412b_map_09fb.ps}%\n  \\hspace{\\stretch{4}}\n\\end{center}\n\\caption[]{Same as Fig.~\\ref{fig:gj51_map} for WX~UMa 2006, 2007, 2008\nand\n2009 data sets (from left to right).}\n\\label{fig:gj412b_map}\n\\end{figure*}\n\nBetween 2006 and 2009, we observed the M6 dwarf WX~UMa during 4 runs, collecting\na total of 29 spectra. The LSD Stokes~$V$ profiles are very similar throughout\nthe data set: a very strong (the peak-to-peak amplitude is close to 2\\% of the\nunpolarised continuum level) simple two-lobbed signature of negative polarity\n(i.e. corresponding to a longitudinal field directed toward the star).  Temporal\nevolution inside each data set, presumably due to rotational modulation, is\nnoticeable though weaker than what we observe on GJ~51. We measure average\nvalues ranging from $\\hbox{${\\rm RV}$}=69.95$ to $70.25~\\hbox{km\\,s$^{-1}$}$ with a jitter ranging from 0.06\nto 0.53~\\hbox{km\\,s$^{-1}$} (depending on the observation epoch, see Tab.~\\ref{tab:obs}).  This\nis in agreement with $\\hbox{${\\rm RV}$}=68.886~\\hbox{km\\,s$^{-1}$}$ \\cite[single precise measurement by ][on\nGJ~412A]{Nidever02}. We use $\\hbox{$v\\sin i$}=5.0~\\hbox{km\\,s$^{-1}$}$ \\citep{Reiners09} which accounts\nfor Zeeman broadening and results in better agreement with our Stokes $I$ and\n$V$ LSD profiles than the previous value of $\\hbox{$v\\sin i$}=7.7~\\hbox{km\\,s$^{-1}$}$ inferred\nfrom correlation profiles \\citep{Delfosse98}. From the circularly\npolarised LSD profiles we infer $\\hbox{$P_{\\rm rot}$}=0.74~\\hbox{$\\rm d$}$. With this rotation\nperiod, our 2007 data cover half of the rotation cycle, and the 3 other\ndata sets result in a good phase coverage. The comparison of the\nresulting $\\hbox{$R\\sin i$}=0.073~\\hbox{${\\rm R}_{\\odot}$}$ with $\\hbox{$R_{\\star}$}=0.12~\\hbox{${\\rm R}_{\\odot}$}$ (from\ntheoretical models) indicates an intermediate inclination angle, we\nset $i=40\\hbox{$^\\circ$}$.\n\n\\begin{table}\n\\begin{center}\n\\caption[]{Same as Tab.~\\ref{tab:gj51_fit} for WX~UMa.} \n\\begin{tabular}{cccccc}\n\\hline\nEpoch & $\\ell_{ZDI}$ & ${\\hbox{$\\chi^2_r$}}_0$ & ${\\hbox{$\\chi^2_r$}}_f$ &\n$\\avg{B}$ & $B_{max}$ \\\\\n & & & & (\\hbox{$\\rm kG$}) & (\\hbox{$\\rm kG$}) \\\\\n\\hline\n2006 & 4 &  8.98 & 0.90 & 0.89 & 3.82 \\\\\n2007 & 4 & 20.48 & 1.05 & 0.94 & 4.88 \\\\\n2008 & 4 & 12.42 & 1.00 & 1.03 & 4.55 \\\\\n2009 & 4 & 14.58 & 1.15 & 1.06 & 4.53 \\\\\n\\hline\n\\label{tab:gj412b_fit}\n\\end{tabular}\n\\end{center}\n\\end{table}\n\nAlthough only the 2006 data set can be fitted below $\\hbox{$\\chi^2_r$}=1.0$ (see\nTab.~\\ref{tab:gj412b_fit}), the ZDI synthetic Stokes~$V$ profiles \nmatch well the evolution of the LSD signatures for all epochs, as shown in\nFig.~\\ref{fig:gj412b_spec}. The corresponding magnetic maps are presented\nin Fig.~\\ref{fig:gj412b_map}: they all feature a strong polar cap of radial\nfield (of negative polarity, i.e.  field lines directed toward the star)\nreaching a maximum flux of approximately 4~\\hbox{$\\rm kG$}, whereas the magnetic flux\naveraged over the visible fraction of the star is about 1~\\hbox{$\\rm kG$}. Azimuthal and\nmeridional field structure are much weaker.  The topology is very simple, modes\nwith degree $\\ell > 4$ can be neglected, the dipole modes encompass more than\n60\\% of the reconstructed energy at all epochs.  Toroidal and non-axisymmetric\ncomponents of the field are very weak. The evolution of the magnetic field over\nsuccessive years is very weak, the maps are strikingly similar.\nThese conclusions are robust to uncertainties on stellar parameters (in\nparticular $i$ and $\\hbox{$v\\sin i$}$). When varying these parameters over the width of\ntheir respective error bars (see section~\\ref{sec:techniques-uncert}) the\nreconstructed magnetic field of WX~UMa is always almost purely poloidal, mostly\naxisymmetric, and the main mode is the radial component of a dipole aligned with\nthe rotation axis. \n\n\\section{DX~Cnc=GJ~1111}\n\\begin{figure*}\n\\begin{center}\n  \\includegraphics[height=0.40\\textheight]{fig_arxiv\/gj1111_spec_070809b.ps}\n\\end{center}\n\\caption[]{Same as Fig.~\\ref{fig:gj51_spec} for DX~Cnc 2007, 2008, and\n2009 data sets (from left to right).}\n\\label{fig:gj1111_spec}\n\\end{figure*}\n\n\\begin{figure*}\n\\begin{center}\n\\includegraphics[height=0.40\\textheight]{fig_arxiv\/gj1111_map_07b.ps}\\hspace{\n0.5cm\n}\n\\includegraphics[height=0.40\\textheight]{fig_arxiv\/gj1111_map_08b.ps}\\hspace{\n0.5cm\n}\n\\includegraphics[height=0.40\\textheight]{fig_arxiv\/gj1111_map_09b.ps}%\n  \\hspace{\\stretch{4}}\n\\end{center}\n\\caption[]{Same as Fig.~\\ref{fig:gj51_map} for DX~Cnc 2007, 2008, and\n2009 data sets (from left to right).}\n\\label{fig:gj1111_map}\n\\end{figure*}\n\nWe carried out 3 observation runs on DX~Cnc between 2007 and 2009, resulting in\n21 pairs of Stokes~$I$ and $V$ spectra. The LSD polarised profiles are\ndisplayed in Fig.~\\ref{fig:gj1111_spec}, the Zeeman signatures have low\namplitudes but are definitely detected in several spectra. The amplitude and\nshape of the circularly polarised line dramatically evolve during each observing\nrun, presumably due to rotational modulation. From the LSD profiles, we measure\naverage RV ranging from $10.44$ to $10.67~\\hbox{km\\,s$^{-1}$}$, with a jitter strongly\ndepending on the observing epoch (see Tab.~\\ref{tab:obs}), in agreement with the\npreviously published values ($\\hbox{${\\rm RV}$}=9~\\hbox{km\\,s$^{-1}$}$ in \\citealt{Delfosse98};\n$\\hbox{${\\rm RV}$}=10.1~\\hbox{km\\,s$^{-1}$}$ in \\citealt{Mohanty03}). We use $\\hbox{$v\\sin i$}=13~\\hbox{km\\,s$^{-1}$}$\n\\cite[][]{Reiners07}, and $\\hbox{$P_{\\rm rot}$}=0.46~\\hbox{$\\rm d$}$ (inferred from our data). As the\nresulting $\\hbox{$R\\sin i$}=0.12~\\hbox{${\\rm R}_{\\odot}$}$ is already higher than $\\hbox{$R_{\\star}$}=0.11~\\hbox{${\\rm R}_{\\odot}$}$\npredicted by theoretical models, we assume a high inclination angle of the\nrotation axis and set $i=60\\hbox{$^\\circ$}$ for the imaging process. \nOur 2007 and 2008 data sets provide a reasonable phase coverage and the 2009 one\nresults in a good sampling of the stellar rotation.\n\n\\begin{table}\n\\begin{center}\n\\caption[]{Same as Tab.~\\ref{tab:gj51_fit} for DX~Cnc.}\n\\begin{tabular}{cccccc}\n\\hline\nEpoch & $\\ell_{ZDI}$ & ${\\hbox{$\\chi^2_r$}}_0$ & ${\\hbox{$\\chi^2_r$}}_f$ &\n$\\avg{B}$ & $B_{max}$ \\\\\n & & & & (\\hbox{$\\rm kG$}) & (\\hbox{$\\rm kG$}) \\\\\n\\hline\n2007 & 6 & 1.70 & 1.00 & 0.11 & 0.22 \\\\\n2008 & 6 & 1.42 & 1.00 & 0.08 & 0.20 \\\\\n2009 & 6 & 1.37 & 1.00 & 0.08 & 0.18 \\\\\n\\hline\n\\label{tab:gj1111_fit}\n\\end{tabular}\n\\end{center}\n\\end{table}\n\nBoth the mean longitudinal field and the standard deviation from this value get\nweaker from one observing run to the next one, indicating intrinsic variability\nof the magnetic field.  Using the ZDI tomographic imaging code, we can fit the 3\ndata sets down to $\\hbox{$\\chi^2_r$}=1.0$, the resulting magnetic fields are presented in\nFig.~\\ref{fig:gj1111_map}. Although for the three epochs, the reconstructed\nmagnetic topologies feature a significant non-axisymmetric component, they\nsignificantly differ from each other: (i) the fraction of magnetic energy\nreconstructed in the toroidal component grows from 7\\% in 2007 to 38\\% in 2009;\n(ii) the two main spots of radial magnetic field seem to evolve between 2007 and\n2008, and finally in 2009 only one region of strong radial field remains. (iii)\nThe averaged magnetic flux decreases from 110~G in 2007 to 80~G in 2008 and\n2009.  This last point is strengthened by the fact that for a given topology,\nZDI recovers less magnetic flux for a dataset providing partial phase coverage\n(due to the maximum entropy constraint), whereas here the larger flux is\ninferred from the dataset providing the poorest sampling of stellar rotation.\nThe presence of a strong toroidal component on DX~Cnc in 2008 and 2009 is a\nrobust result. In particular, when varying the input parameters within their\nrespective error bars (see Sec.~\\ref{sec:techniques-uncert}) the toroidal\ncomponent always accounts for at least 30~\\% of the reconstructed magnetic\nenergy.\n\n\\section{GJ~3622}\n\n\\begin{figure*}\n\\begin{center}\n  \\includegraphics[height=0.40\\textheight]{fig_arxiv\/gj3622_spec_0809a.ps}%\n  \\hspace{0.5cm}%\n  \\includegraphics[height=0.40\\textheight]{fig_arxiv\/gj3622_map_08a.ps}%\n  \\hspace{0.5cm}%\n  \\includegraphics[height=0.40\\textheight]{fig_arxiv\/gj3622_map_09a.ps}%\n\\end{center}\n\\caption[]{2008 and 2009 spectra and magnetic maps of GJ~3622. See\n  Fig.~\\ref{fig:gj51_spec} and \\ref{fig:gj51_map} for more details.}\n\\label{fig:gj3622}\n\\end{figure*}\n\nGJ~3622 was observed in 2008 and 2009, we collected 8 and 5\nsequences, respectively. In spite of relatively low \\hbox{S\/N}\\ (due to a low\nintrinsic luminosity), Stokes $V$ signatures are clearly detected in\nseveral spectra and variations are undoubtedly noticeable (see\nFig.~\\ref{fig:gj3622}). From our data sets we infer $\\hbox{$P_{\\rm rot}$}=1.5~\\hbox{$\\rm d$}$,\nand use $\\hbox{$v\\sin i$}=3~\\hbox{km\\,s$^{-1}$}$ reported by \\cite{Mohanty03}, resulting in\n$\\hbox{$R\\sin i$}=0.09~\\hbox{${\\rm R}_{\\odot}$}$. Our data sets provide a reasonable phase coverage with\nthis period. As evolutionary models predict $\\hbox{$R_{\\star}$}=0.11~\\hbox{${\\rm R}_{\\odot}$}$\nfor a $0.09~\\hbox{${\\rm M}_{\\odot}$}$ M dwarf, we set the inclination angle to 60\\hbox{$^\\circ$}\\ for\nthe imaging process.\n\nSetting $\\ell_{max}=2$, given the very weak signal detected, it is\npossible to fit our 2 data sets down to noise level (see\nTab.~\\ref{tab:gj3622_fit}). The corresponding magnetic maps (see\nFig.~\\ref{fig:gj3622}) are very similar. A radial field spot of negative\npolarity, i.e. field lines directed toward the star, is located at\nmid-latitudes. Magnetic flux reaches up to 110~G in\nthis region. Azimuthal and meridional fields are much weaker, the\ntoroidal component represents less than 10~\\% of the overall magnetic\nenergy at both epochs. The reconstructed topology is close to a tilted\ndipole: less than 10~\\% of the magnetic energy is reconstructed in\n$\\ell=2$ modes, and the axisymmetric component stands for more than\n80~\\% of the energy content. The data being very noisy the simple topology\nreconstructed by ZDI likely reflects the lack of information in the\npolarised spectra. \n\n\\begin{table}\n\\begin{center}\n\\caption[]{Same as Tab.~\\ref{tab:gj51_fit} for GJ~3622.}\n\\begin{tabular}{cccccc}\n\\hline\nEpoch & $\\ell_{ZDI}$ & ${\\hbox{$\\chi^2_r$}}_0$ & ${\\hbox{$\\chi^2_r$}}_f$ &\n$\\avg{B}$ & $B_{max}$ \\\\\n & & & & (\\hbox{$\\rm kG$}) & (\\hbox{$\\rm kG$}) \\\\\n\\hline\n2008 & 2 & 2.08 & 0.95 & 0.05 & 0.11 \\\\\n2009 & 2 & 1.68 & 0.95 & 0.06 & 0.11 \\\\\n\\hline\n\\label{tab:gj3622_fit}\n\\end{tabular}\n\\end{center}\n\\end{table}\n\n\\section{other stars}\n\\label{sec:other}\n\n \\begin{figure*}\n   \\centering\n  \\includegraphics[width=0.32\\textwidth]{fig_arxiv\/gj1154a_IV.ps} %\n  \\includegraphics[width=0.32\\textwidth]{fig_arxiv\/gj1224_IV.ps} %\n  \\includegraphics[width=0.32\\textwidth]{fig_arxiv\/gl406_IV.ps} \n  \\caption[]{Stokes~$I$ (lower panels) and $V$ (upper panels) LSD\nsignatures of\nGJ~1154~A, GJ~1224 and CN~Leo, from left to right. In each panel all the\nprofiles of the time series are plotted as superimposed grey lines, and the\naverage profile is shown in red. Vertical dotted lines represent the line center\n(in bold) and the approximate limits of the line. In Stokes~$V$ plots $\\pm\n1\\sigma$ levels (corresponding the individual spectra) are shown as dashed\nlines, and the reference level as a dotted line.} \\label{fig:3nozdi}\n\\end{figure*}\n\nFor five stars of our sample we collected time series of polarised spectra but\ncould not produce a definitive magnetic map. For GJ~1154~A, GJ~1224 and CN~Leo\n(GJ~406) we detect very strong and simple signatures (see\nFig.~\\ref{fig:3nozdi}). To our knowledge, no rotation periods have been measured\nfor these stars and our data sets do not allow us to conclude, either because of\nlow intrinsic variability of the Zeeman signature or poor phase\nsampling. Although we cannot compute a magnetic map for these stars, the\ncollected spectra unmistakably show that they host very strong\nlarge-scale magnetic fields (longitudinal fields are about 600~G).\nStrong magnetic fields\n(total magnetic fluxes in the 2-3~\\hbox{$\\rm kG$}\\ range) have been previously detected\non these stars by \\cite{Reiners07} and \\cite{Reiners09} from the analysis of\nunpolarised spectra (see Tab.~\\ref{tab:sample}). The simple signatures\n(two-lobbed antisymmetric) featuring very low variability also clearly\nsuggest that these fields are mostly poloidal, strongly axisymmetric and\npresumably dominated by low degree modes, similar to what we observe on\nGJ~51 and WX~UMa for instance. The low dispersions of longitudinal\nfields and RV values in each data set may indicate that phase sampling\nis loose and thus rotation period close to a fraction of day, or\/and\nthat these stars are observed nearly pole-on.\n\nFor the faintest stars of the sample VB~8 (GJ~644~C) and VB~10 (GJ~752~B), the\nStokes~$V$ signatures are too weak to be definitely detected in individual LSD\nspectra. The initial ${\\hbox{$\\chi^2_r$}}_0$ are respectively equal to 1.089 and 1.150.\nLSD signatures observed on these stars are shown on Figure~\\ref{fig:vb8-10}. By\naveraging all the LSD profiles of a data set, the noise level is decreased but\nonly features visible on all spectra --- corresponding to the axisymmetric\ncomponent of the field, if rotation sampling is even --- remain visible.  The\nresulting signal shown in Fig.~\\ref{fig:vb8-10} (bold red line) is processed\nwith a zero phase shift low-pass filter to remove the frequencies higher than\npermitted by the instrumental profile (width of 4.6~\\hbox{km\\,s$^{-1}$}, or 2.5 LSD pixels).\nFor VB~8, the averaged LSD profile does not feature any significant signal\n(\\hbox{$\\chi^2_r$}=0.99), indicating that the axisymmetric component of the magnetic field\nof this star is too weak to be detected. The averaged profile of VB~10 features\na weak but distinguishable signature corresponding to $\\hbox{$\\chi^2_r$} = 1.90$.  It\nsuggests the presence of a large-scale magnetic field having a\nsignificant axisymmetric component, but further observations are needed to\nconfirm this point. From observations in unpolarised\nlight, \\cite{Reiners07} report total magnetic fluxes of 2.3 and 1.3~\\hbox{$\\rm kG$}\\ on VB~8\nand VB~10, respectively (see Tab.~\\ref{tab:sample}). The very weak\nStokes~$V$ signatures observed here (with corresponding maximum longitudinal\nfields of the order of 100~G) suggest that the magnetic field of these stars\nis mainly structured on small spatial scales.\n\n\\begin{figure*}\n  \\centering\n  \\includegraphics[width=0.45\\textwidth]{fig_arxiv\/vb8_IV.ps} %\n  \\includegraphics[width=0.45\\textwidth]{fig_arxiv\/vb10_IV.ps} %\n  \\caption[]{Same as Fig.~\\ref{fig:3nozdi} for VB~8 and VB~10. The bold\nred line results from low-pass filtering of the Stokes~$V$ signatures and\ndash-dotted lines show the $\\pm 1\\sigma$ levels corresponding the averaged\nspectra.}\n  \\label{fig:vb8-10}\n\\end{figure*}\n\nPerforming a ZDI analysis on the VB~10 time series, we find 2 possible rotation\nperiods: 0.52, and  0.69~\\hbox{$\\rm d$}, the second being favored by photometric\nmeasurements (MEarth project, J.~Irwin, private communication).\nFigure~\\ref{fig:vb10_spec_maps} shows the fit achieved for $\\hbox{$P_{\\rm rot}$}=0.69~\\hbox{$\\rm d$}$,\n$i=60~\\hbox{$^\\circ$}$ and the corresponding magnetic maps. As expected from the signature\nshape (non antisymmetric with respect to the line centre), the reconstructed\nmagnetic field exhibits a significant axisymmetric toroidal component. A\nnon-axisymmetric poloidal field (tilted quadrupole, mode $\\alpha_{22}$) is also\nreconstructed to fit the Stokes~$V$ component that varies along rotation.\n\n\\begin{figure*}\n  \\centering\n  \\includegraphics[height=0.4\\textheight]{fig_arxiv\/vb10_spec_09f.ps}\n  \\includegraphics[height=0.4\\textheight]{fig_arxiv\/vb10_map_09f.ps}\n  \\caption[]{2009 spectra and magnetic maps of VB~10. See\n  Fig.~\\ref{fig:gj51_spec} and \\ref{fig:gj51_map} for more details. Data\nare\n  phased according to the ephemeris ${\\rm HJD}=2\\,455\\,000 + 0.69\\,E$.}\n  \\label{fig:vb10_spec_maps}\n\\end{figure*}\n\n\\section{Discussion and conclusion}\n\\label{sec:disc}\n\\begin{table*}\n\\caption[]{Magnetic quantities derived from our study. For each star,\ndifferent observation epochs are presented separately. In columns 2--4 we report\nquantities from Table~\\ref{tab:sample}, respectively the stellar mass, the\nrotation period, and the effective Rossby number. Columns 5, 6 and 7\nmention the Stokes $V$ filling factor, the reconstructed magnetic energy and the\naverage magnetic flux. Columns 8--11 list the percentage of reconstructed\nmagnetic energy respectively lying in poloidal, dipole (poloidal and $\\ell=1$),\nquadrupole (poloidal and $\\ell=2$), and octupole (poloidal and $\\ell=3$) modes.\nIn column 12, we mention the percentage of magnetic energy reconstructed\nin axisymmetric modes (defined as $m < \\ell\/2$) and the percentage of\npoloidal energy in axisymmetric modes.\nSee section~\\ref{sec:techniques-uncert} for a discussion on the robustness of\nmagnetic map reconstruction and the uncertainties\nassociated with the derived quantities.\n}\n \\begin{tabular}{ccccccccccccc}\n\\hline\nName & Mass & \\hbox{$P_{\\rm rot}$} & $Ro$ & $f_V$ & $\\avg{B^2}$ & $\\avg{B}$ & pol. &\ndipole &\nquad. & oct. & axisymm. \\\\\n & (\\hbox{${\\rm M}_{\\odot}$}) & (d) & ($10^{-2}$) & & ($\\rm10^5\\,G^2$) & (kG) & (\\%) &\n(\\%) & (\\%) & (\\%) & (\\%) \\\\ \n\\hline\nGJ~51$^1$ (06) & 0.21 & 1.02 & 1.2 & 0.12 & 38.6 & 1.61 & 99 & 96 &\n0 & 2 & 91\/91\\\\\n\\phantom{GJ~51} (07) &--&--&--& 0.12 & 31.3 & 1.58 & 99 & 92 & 0 & 6 &\n77\/77\\\\ \n\\phantom{GJ~51} (08) &--&--&--& 0.12 & 32.6 & 1.65 & 97 & 92 & 1 & 3 &\n89\/89\\\\ \nGJ~1156$^2$ (07) & 0.14 & 0.49 & 0.5 & 1.0 & 0.06 & 0.05 & 88 & 30 & 26 & 19 &\n6\/3\\\\\n\\phantom{GJ~1156} (08) &--&--&--& 1.0 & 0.19 & 0.11 & 83 & 41 & 28 & 11 &\n20\/12\\\\\n\\phantom{GJ~1156} (09) &--&--&--& 1.0 & 0.13 & 0.10 & 94 & 54 & 24 & 10 &\n2\/1\\\\\nGJ~1245~B (06) & 0.12 & 0.71 & 0.7 & 0.06 & 0.44 & 0.17 & 80 & 45 & 14 & 13 &\n15\/9\\\\\n\\phantom{GJ~1245~B} (07) &--&--&--& 0.10 & 0.49& 0.18 & 84 & 46 & 27 & 7 &\n52\/53\\\\\n\\phantom{GJ~1245~B} (08) &--&--&--& 0.10 & 0.06 & 0.06 & 85 & 33 & 25 & 19 &\n20\/18\\\\\nWX~UMa (06) & 0.10 & 0.78 & 0.8 & 0.12 & 16.08 & 0.89 & 97 & 66 & 21 & 6 &\n92\/92\\\\\n\\phantom{WX~UMa} (07) &--&--&--& 0.12 & 24.42 & 0.94 & 97 & 71 & 13 & 3 &\n92\/94\\\\\n\\phantom{WX~UMa} (08) &--&--&--& 0.12 & 23.53 & 1.03 & 97 & 69 & 19 & 6 &\n83\/85\\\\\n\\phantom{WX~UMa} (09) &--&--&--& 0.12 & 37.54 & 1.06 & 96 & 89 & 2 & 2 &\n95\/96\\\\\nDX~Cnc (07) & 0.10 & 0.46 & 0.5 & 0.20 & 0.17 & 0.11 & 93 & 69 & 11 & 9 &\n77\/77\\\\\n\\phantom{DX~Cnc} (08) &--&--&--& 0.20 & 0.09 & 0.08 & 73 & 31 & 25 & 10 &\n49\/34\\\\\n\\phantom{DX~Cnc} (09) &--&--&--& 0.20 & 0.09 & 0.08 & 62 & 42 & 11 & 4 &\n70\/61\\\\\nGJ~3622 (08) & 0.09 & 1.5 & 1.5 & 1.0 & 0.04 & 0.05 & 96 & 90 & 7 & -- &\n73\/72\\\\ \n\\phantom{GJ~3622} (09) &--&--&--& 1.0 & 0.05 & 0.06 & 93 & 84 & 9 & -- & 80\/78\\\\\n\\hline\n \\label{tab:syn}\n \\end{tabular}\n \\begin{flushleft}\n   $^1$ For GJ~51, the imaging process is weakly constrained by our\n   data sets due to poor phase coverage (see Sec.~\\ref{sec:gj51}).\\\\\n   $^2$ For GJ~1156 the alternative rotation period 0.33~d cannot be\n   definitely excluded (see Sec.~\\ref{sec:gj1156}). In this case, the\n   reconstructed topologies remain   similar for the 3 epochs, and our\n   conclusions are not affected. \n \\end{flushleft}\n\\end{table*}\n\nWe present the final part of our exploratory spectropolarimetric\nsurvey of M dwarfs, following D08 and M08b (respectively concentrating on\nmid and early M dwarfs) we focus here on the low mass end of our sample.\nFor 6 stars, it is possible to apply ZDI techniques to the time-series\nof circularly polarised spectra and thus to infer the large-scale component of\ntheir magnetic topologies. The properties of the reconstructed topologies of\nthese stars are presented in Table~\\ref{tab:syn}. For the remaining 5 objects,\nthe data sets do not permit such a study, it is however possible to retrieve\nsome constraints about their magnetic properties.\n\nTwo stars of the subsample (namely GJ~51 and WX~UMa) exhibit large-scale\nmagnetic fields very similar to those observed by M08b on mid M dwarfs,\ni.e. very strong, axisymmetric poloidal and nearly dipolar fields, with\nvery little temporal variations. For three stars for which we cannot perform a\ndefinitive ZDI reconstruction (GJ~1154~A, GJ~1224 and CN~Leo) our data\nstrongly suggest similar topologies. From the observations of WX~UMa, we\ndemonstrate that the timescale of temporal evolution in the magnetic topologies\nof these stars can be larger than 3~years, whereas previous observations by M08a\nand M08b were based on observations spanning only 1~year.\n\nThe other stars for which we can reconstruct the large-scale magnetic\ntopologies, are clearly different. They are weaker than those of the first\ncategory, and generally feature a significant non-axisymmetric\ncomponent, plus a significant toroidal component (although the field is always\npredominantly poloidal).\nTemporal variability is also noticeable, in particular\nfor GJ~1245~B our data indicate unambiguously that the magnetic field strength\nhas dramatically decreased between our 2007 and 2008 observations.\nOur conclusions are robust to uncertainties on stellar parameters\n(\\hbox{$P_{\\rm rot}$}, $i$ and $\\hbox{$v\\sin i$}$). When varying these parameters over the width\nof their respective error bars (see section~\\ref{sec:techniques-uncert}), the\nmain properties of the reconstructed magnetic topologies remain unchanged.\n\n\\begin{figure*}\n  \\centering\n  \\includegraphics[height=1.0\\textwidth, angle=270]{fig_arxiv\/plotMP.ps}\n  \\caption{Properties of the magnetic topologies of our sample of M dwarfs as a\n  function of rotation period and stellar mass. Larger symbols indicate larger\n  magnetic fields while symbol shapes depict the different degrees of\n  axisymmetry of the\n  reconstructed magnetic field (from decagons for purely axisymmetric fields to\n  sharp stars for purely non axisymmetric fields). Colours illustrate the field\n  configuration (dark blue for purely toroidal fields, dark red for purely\n  poloidal fields and intermediate colours for intermediate configurations).\n  Solid lines represent contours of constant Rossby number $Ro=0.1$ and $0.01$\n  respectively corresponding approximately to the saturation and\n  super-saturation thresholds \\citep[e.g.,][]{Pizzolato03}. The theoretical\n  full-convection limit ($\\hbox{$M_{\\star}$} \\simeq0.35\\hbox{${\\rm M}_{\\odot}$}$, \\citealt{Chabrier97}) is\n  plotted as a horizontal dashed line, and the approximate limits of the three\n  stellar groups  discussed in the text are represented as horizontal solid\n  lines. Stars with  $\\hbox{$M_{\\star}$} > 0.45~\\hbox{${\\rm M}_{\\odot}$}$ are from\n  D08, whereas those with $0.25<\\hbox{$M_{\\star}$}<0.45~\\hbox{${\\rm M}_{\\odot}$}$ are from M08b. For GJ~1245~B\n  symbols corresponding to 2007 and 2008 data sets are superimposed in order to\n  emphasize the variability of this object. %\n  Uncertainties associated with the plotted magnetic quantities are discussed\n  in section~\\ref{sec:techniques-uncert}.}\n\\label{fig:plotMP}\n\\end{figure*}\n\nThese results are presented in a more visual way in Fig.~\\ref{fig:plotMP}.\nPrevious studies by D08 and M08b have revealed strong evidence that a clear\ntransition occurs at approximately\n0.5~\\hbox{${\\rm M}_{\\odot}$}\\, \\ie more or less coincident with the transition to a\nfully convective internal structure. The situation here is different, we find\nstars with similar stellar parameters that exhibit radically different magnetic\ntopologies. On Fig.~\\ref{fig:plotMP}, WX~UMa is the only star below 0.2~\\hbox{${\\rm M}_{\\odot}$}\\\nto host a mid-M-dwarf-like field. Whereas DX~Cnc and GJ~1245~B are very\nclose to it in the mass-rotation plane they feature fields with very different\nproperties. This observation may be explained in several ways. For instance,\nanother parameter than mass and rotation period, such as stellar age, may play a\nrole. In our sample we indeed notice that most stars below 0.15~\\hbox{${\\rm M}_{\\odot}$}\\ that\nexhibit a weak complex field belong to a young kinematic population\naccording to \\citealt{Delfosse98} (GJ~1156, DX~Cnc and GJ~3622), whereas those\nhosting a strong dipolar field (WX~UMa, GJ~1224 and CN~Leo) belong to older\nkinematic populations (old disk and old\/young disk). This hypothesis requires\nfurther investigation. One could also imagine, for instance, that the\nmagnetic fields of very low mass stars may switch between two different states\nover time. This hypothesis is supported by the fact that for one of the stars in\nthe weak and complex field regime (GJ~1245~B) we observe a dramatic variation of\nthe magnetic flux on a timescale of one year which may indicate that the\nmagnetic field of these objects may go through chaotic variations and eventually\nswitch between the two categories of field actually observed. However no such\nswitch has been observed in our sample. We observe 5 objects in the strong\ndipole field category (GJ~51, GJ~1154~A, GJ~1224, CN~Leo and WX~UMa) and 6 in\nthe weak field category (GJ~1156, GJ~1245~B, DX~Cnc, GJ~3622, VB~8 and VB~10),\nindicating that stars would spend as much time in both states. No star is\nobserved in an intermediate state, suggesting that a putative transition would\nbe fast. This hypothesis may be investigated through the analysis of long-term\nradio monitoring, since radio emission would be presumably strongly impacted by\nsuch a dramatic change of the stellar magnetic field. Recent observations of\nultracool dwarfs reveal a long-term variability of the activity indices\n\\cite[\\eg][]{Antonova07, Berger10} that may support this view.\n\n\\begin{figure*}\n \\centering\n \\includegraphics[angle=270, width=0.50\\textwidth]{fig_arxiv\/Ro_Bf1b.ps}%\n \\includegraphics[angle=270, width=0.50\\textwidth]{fig_arxiv\/Ro_Bf2b.ps}%\n \\caption[]{Magnetic flux as a function of Rossby number. On the left\npanel magnetic fluxes as measured from Stokes~$V$ spectra and ZDI by M08b\nand D08 are mentioned as blue hexagons ($\\hbox{$M_{\\star}$}>0.4~\\hbox{${\\rm M}_{\\odot}$}$) and green\nsquares ($0.2<\\hbox{$M_{\\star}$}<0.4~\\hbox{${\\rm M}_{\\odot}$}$). Results from this paper\n($\\hbox{$M_{\\star}$}\\leq0.2~\\hbox{${\\rm M}_{\\odot}$}$) are shown as red circles. On the\nright panel the ratio between the magnetic fluxes as recovered from Stokes~$V$\nand Stokes~$I$ measurements (whenever available) are shown. Stokes~$I$ magnetic\nfluxes are taken from \\cite{Johns00}, \\cite{Reiners07}, \\cite{Reiners09}, and\n\\cite{Reiners09b} (see Tab.~\\ref{tab:sample}). Measurements of different epochs\n(whenever available) for Stokes~$V$ are shown connected by a solid line. On\neach plot the $Ro=10^{-1}$, corresponding to the saturation level, is depicted\nas a vertical solid black line. Triangles denote upper limits. In the left\npanel, horizontal lines show the magnetic fluxes corresponding to saturation for\nstars with $\\hbox{$M_{\\star}$}>0.4~\\hbox{${\\rm M}_{\\odot}$}$ (blue), and $0.2<\\hbox{$M_{\\star}$}<0.4~\\hbox{${\\rm M}_{\\odot}$}$ (green). In\nthe right panel they show the mean fraction of magnetic flux detected in\nStokes~$V$ spectra.}\n \\label{fig:RoBsq}\n\\end{figure*}\n\nOn the left panel of Fig.~\\ref{fig:RoBsq}, we plot the reconstructed magnetic\nflux as a function of the Rossby number. As mentioned in D08, stars with\n$\\hbox{$M_{\\star}$} > 0.4~\\hbox{${\\rm M}_{\\odot}$}$ follow the expected rotation--magnetic field connection,\nwith saturation for $Ro \\lesssim 0.2$. Whereas for $\\hbox{$M_{\\star}$} > 0.4~\\hbox{${\\rm M}_{\\odot}$}$, we only\nobserve objects in the saturated regime, with a significantly stronger\nsaturation magnetic flux. All the stars of the late M subsample have Rossby\nnumbers below $2\\times10^{-2}$ and are thus expected to be in the saturated\nregime. As in Fig.~\\ref{fig:plotMP} the stars studied here can be divided into\ntwo distinct categories. The first one is composed of stars hosting a very\nstrong magnetic field that lie in the saturated part of the rotation-activity\nrelation, similar to mid M dwarfs. We notice that significantly higher magnetic\nfluxes are reconstructed for WX~UMa and GJ~51 than for mid M dwarfs studied by\nM08b, although all these stars seem to lie in the saturated dynamo regime. For\nthe second category of late M dwarfs (having a weak complex field), we recover\nless magnetic flux (even less than for some early M dwarfs studied by D08), in\nspite of their very low Rossby number. Super-saturation is unlikely to be the\nexplanation since stars of both categories have similar Rossby numbers. As\nmentioned for Fig.~\\ref{fig:plotMP}, we observe no object in an intermediate\nstate. The aforementioned variability of GJ~1245~B is also prominent in this\nplot.\n\nOn the right panel of Fig.~\\ref{fig:RoBsq}, magnetic flux inferred from\nStokes~$V$ measurements are compared to those derived from Stokes~$I$.\nAgain, stars with masses below and above 0.4~\\hbox{${\\rm M}_{\\odot}$}\\ studied by\nD08 and M08b clearly form two separate groups. In partly-convective\nstars only a few percents of the magnetic flux measured in $I$ is detectable in\n$V$ similarly to what is observed for the Sun, while this ratio is\nclose to 15\\% in fully convective ones. This indicates a higher degree of\norganization of the field in fully convective mid M dwarfs, with more magnetic\nflux in the spherical harmonics of lowest degree. The ratios of about 30\\%\nplotted for WX~UMa are upper limits (since the flux based on Stokes~$I$ is a\nlower limit), it is thus not clear if this star differs from mid M dwarfs in\nthis respect. This high value however indicates a high degree of organization.\nFor DX~Cnc, GJ~1245~B and GJ~1156, the ratio of Stokes~$V$ and $I$ fluxes is\ncloser to the early M dwarfs value. Therefore the magnetic fields of the weak\nfield category of late M dwarfs share similar properties with that of partly\nconvective M dwarfs. \n\nThese results on the magnetic topologies of M dwarfs also suggest that\ndynamo processes in low-mass main sequence stars and pre-main sequence stars\nmay be similar. Indeed the first spectropolarimetric results on young stars show\nthat the fully convective T Tauri star BP~Tau (0.7~\\hbox{${\\rm M}_{\\odot}$}) exhibit a strong\nlarge-scale magnetic field \\cite[][]{Donati08a}, whereas the more massive partly\nconvective star V2129~Oph (1.4~\\hbox{${\\rm M}_{\\odot}$}) possesses a weaker and more complex\nfield \\cite[][]{Donati07}. This is reminiscent of the transition we observe at\n0.5~\\hbox{${\\rm M}_{\\odot}$}\\ among main sequence M dwarfs. Recent observations, on the fully\nconvective V2247~Oph (0.35~\\hbox{${\\rm M}_{\\odot}$}) reveal a still weaker and more complex\nmagnetic field \\cite[][]{Donati10} which may correspond to the weak field late M\ndwarf we present here. \n\nWe do not detect Stokes~$V$ signatures in individual spectra of the two latest\nstars of our sample VB~8 and VB~10. However the averaged LSD profile of VB~10\nsuggests the presence of a toroidal axisymmetric field component on this object.\nFurther observations may confirm this first spectropolarimetric detection on an\nultra-cool dwarf.\n\nThe first spectropolarimetric survey of M dwarfs has already provided dynamo\ntheorists with strong constraints on the evolution of surface magnetic fields of\nM dwarfs across the fully convective divide (D08 and M08b). The results\npresented in this paper on the magnetic topologies of late M dwarfs reveal a new\nunexpected behaviour below 0.2~\\hbox{${\\rm M}_{\\odot}$}. We interpret the fact that objects with\nsimilar stellar parameters host radically different magnetic topologies as a\npossible evidence for a switch between two dynamo states (either cyclic or\nchaotic). Finally our observations suggest the presence of a large-scale\nmagnetic field on the M8 dwarf VB~10, featuring a significant toroidal\naxisymmetric component, whereas the detection of the magnetic field of VB~8 (M7)\nis not possible from our spectropolarimetric data.\n\n\\section*{ACKNOWLEDGEMENTS} \nThe authors thank the CFHT staff for their valuable help throughout our\nobserving runs. We are grateful to Jonathan Irwin and Joel Hartman for\nproviding results prior to publication on the photometric periods of the M\ndwarfs studied in this paper. We also thank the referee Gibor Basri for his\nfruitful suggestions.\n\n\n\\bibliographystyle{mn2e}\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
{"text":"\\section{\\sc \\bf Introduction}\n\\label{sec:1}\n\nTopological singularities such as points, lines and walls are ubiquitous in phases with broken symmetry. \nCanonical examples include dislocations in solids\\cite{klelav}, vortex lines and rings\\cite{rututu} in \nsuperfluid ${}^3He$ and ${}^4He$, Abrikosov vortex lines in superconductors\\cite{bigahumur}, vortex \nlines in Bose-Einstein condensates\\cite{weneschbrdaan}, umbilic lines\\cite{macgar} and $\\pi$ \nsolitons\\cite{veinblsmlavnob} (disclinations) in nematic liquid crystals (NLC) that provide a testing \nground for theories of cosmology\\cite{chdutuyur,rututu}, $\\lambda$ lines in cholesteric fluids\\cite{klefrie}, \nBloch and N\\'{e}el lines in ferromagnets\\cite{kleman}, walls in lipid membranes\\cite{drbrjoad} and \nstring networks in ecology\\cite{avbamenoli}. NLC phases display rich birefringence under a polarizing \nmicroscope during phase ordering from a disordered state after a rapid quench in pressure or temperature, \nresulting in the formation of disclinations with integer and fractional topological charge. These \nsingularities proliferate after nucleation and form contractile loops after intercommutation\\cite{mermin}. \nUnlike dislocations, disclinations possess intricate kinetics, microstructure, and equivalence with \nan electric charge. Strings are charge neutral with either topological charge $\\pm1$ or $\\pm1\/2$ \nresiding at the two segments or end points of the string to form topological dipoles. Higher multipoles \nand integer charged dipoles also nucleate within the charge neutral strings at the early stage of \nkinetics. Subsequently, these structures rupture into fractionally charged dipolar strings. Similar \nto electrodynamics, like topological charges repel and unlike charges attract and annihilate in pairs \nwhile monopoles are nonexistent to retain charge neutrality unless created by symmetry-breaking \nboundaries, an inclusion of impurity or external drive with a laser beam\\cite{niskcorazumu}. \n\nExistence, classification and recombination rules of disclinations in equilibrium, which play an \nimportant role in the material design, is governed by the energy landscape as well as the geometry \n(topology) of the order parameter space\\cite{mermin}. Strings in uniaxial NLC displayed in figure\n\\ref{fig:1} (frames {\\bf a-d}) are topologically defined by $\\pi_1(\\mathcal{RP}_2)=\\mathbb{Z}_2$ \nwith homotopy group $\\pi_1$ in the projective plane $\\mathcal{RP}_2$ resulting in the abelian group \n$\\mathbb{Z}_2$ with topological charge $\\pm1\/2$\\cite{mermin}. After a theoretical proposal\\cite{frank}, \n$\\pi$ solitons have been seen in fluorescence confocal polarized-light microscopy of pentylcyanobiphenyl \n(5CB) NLC\\cite{chdutuyur,veinblsmlavnob}, molecular simulations\\cite{bismallopel} and field theoretic \ncomputations\\cite{bhatam,niskcorazumu}. Likewise, biaxial disclinations displayed in figure \\ref{fig:1} \n(frames {\\bf e-h}) are defined by $\\pi_1(\\mathcal{RP}_3)= \\mathbb{Q}_8$ where $\\mathcal{RP}_3$ is the \nprojective plane and $\\mathbb{Q}_8$ is the nonabelian group of quaternions generating three classes \nof half-integer topological charges denoted by $C_{x,y,z}$. In monolayered thin films, the simultaneous \nand pairwise coexistence of fractionally charged point dipoles of either class $C_{x,y},C_{y,z}$ or \n$C_{x,z}$ is predicted\\cite{kobtho} and observed in field theoretic computations\\cite{zapgolgol,bhatam}. \nAlbeit topologically proscribed in three dimensions, strings of disparate topology do not entangle but \nannihilate pairwise within the respective class\\cite{bhatam}.\n\\begin{figure*}\n\\centering\n\\includegraphics[width=0.75\\textwidth, height=0.6\\textwidth]{figure1.pdf}\n\\caption{{\\bf a,} $\\pi$ solitons in a thick uniaxial film of $5CB$ after a temperature quench from an isotropic \nstate ($T=40^{\\circ}$C) to a nematic state ($T=33.65^{\\circ}$C). Supercooling and superheating temperatures \nare $T=\\{34.2,34.47\\}^{\\circ}$C. Disclination isosurfaces correspond to scalar uniaxial order \nwith isovalue $S_{ueq}\/2$, where $S_{ueq}=0.086$. {\\bf b,} Corresponding biaxial order with isovalue \n$(B_2)_{max}\/2$ where $(B_2)_{max}=0.05$. {\\bf c,} uniaxial order and director distribution on a portion \nof the ${\\it xy}$-slice plane of ({\\bf a}). {\\bf d,} Corresponding spatial extension of scalar uniaxial \nand biaxial order, displaying core structure of two segments of charge neutral disclinations in-plane, \nwhich are $\\pm1\/2$ integer defects\\cite{veinblsmlavnob,niskcorazumu}. {\\bf e,} Charge neutral $\\pi$ \nsolitons of different homotopy class in thermotropic biaxial media (see Supplementary Movie S2 and \nAppendix (section \\ref{app:3}) for defect characterization scheme). Disclination isosurfaces correspond to isovalue $6S_{beq}\/7$ \nwhere $S_{beq}=0.96$. {\\bf f,} Corresponding biaxial order with isovalues \\{$0.42(B_2)_{beq}$,$0.83(B_2)_{beq}$\\} \nfor homotopy class $\\{C_y,C_z\\}$ where $(B_2)_{beq}=1.2$. {\\bf g,} Uniaxial order and director distribution \non a portion of the ${\\it xy}$-slice plane of ({\\bf e}). Note the similarity in the microstructure of \n$\\pm1\/2\\;C_z$ defects with uniaxial defects in ({\\bf c}) and no variation in {\\bf n} for $\\pm1\/2\\;C_y$ \ndefects. {\\bf h,} The corresponding variation of uniaxial and biaxial order displaying core structure \nof disclination segment of both class in-plane. Although both scalar orders decrease in the core of \n$C_y$ defect, biaxiality increases for a decreasing uniaxiality in $C_z$ defect. Parameters are defined \nin Appendix (section \\ref{app:1}) and material (computation) parameters are tabulated in Table \\ref{tbl:GLdGparam}.} \n\\label{fig:1}\n\\end{figure*}\nDepending on the anisotropic elastic constants of the medium, soft disclinations are vulnerable to \nthermal fluctuations and external stimulus like an electromagnetic field. Regulated by the sign of \nthe dielectric anisotropy constant of the material, an electric field at the Fr{\\'e}edericksz threshold \ncan orient the nematic director along or perpendicular to the field direction. It is particularly \ninteresting to examine whether locally uniform and nonuniform electric field can lead to a time-dilated \nkinetics of the disclination network\\cite{niskcorazumu}, and, how the anisotropy of the nematic \norientation embedded in the dielectric tensor leads to nonuniformity in the local electric field.\nFor example, nematic regions at the top of a colloidal inclusion are generated due to the asymmetric \ndistribution of the field intensity\\cite{ucaronu}. Such control is hard to characterize in experiments, \nimpossible in nanoscale molecular simulations and limited in field theoretic calculations due to \nnumerical complexity, as Ref.\\cite{ucaronu} mentions, ``molecular alignment in the inhomogeneous \nelectric field has not yet been well studied as it is not easy to solve the Poisson equation with \nan inhomogeneous dielectric constant to calculate the local electric field''. Attributing to the \nscale invariant property of the Ginzburg-Landau-de Gennes (GLdG) field theory, relaxational kinetics \nof the orientation tensor has quantitatively reproduced experiments {\\it in silico} from \nmesoscale\\cite{bhatnucl,niskcorazumu} to nanoscale\\cite{winalej}. {\\it State of the art} grand \nchallenge is attributed to the nonavailability of a robust numerical scheme\\cite{abukhdeir} which \nis, in descending order of complexity, (a) free from numerical artifacts of the traditional \nmethods\\cite{todeyeo}, accounts for (b) local nonuniformity in electric field and (c) equilibrium \nthermal fluctuations by respecting physical laws, (d) guarantees zero-trace property of the \norientation tensor and (e) incorporates anisotropic elasticity to probe beyond the single diffusion \n(one elastic constant) approximation\\cite{todeyeo}. Recent advances in fluctuating hydrodynamics of \nisotropic suspensions\\cite{donobhgabe} incorporating point (a) demand a natural, yet challenging, \nextension for anisotropic suspensions\\cite{leonar} while on the other hand, numerical achievement \nof points (c-e) is fairly recent\\cite{bhmeads,bhatnucl}.\n\\begin{table*}\n\\small\n\\begin{tabular*}{1.0\\textwidth}{@{\\extracolsep{\\fill}}||c||c||c||c||c||c||c||c||c||c||c||c|}\n\\hline\nFig. & $\\Gamma (P^{-1})$ & $A (Jcm^{-3})$ & $B(Jcm^{-3})$ & $C(Jcm^{-3})$ & $E^\\prime(Jcm^{-3})$ \n&$L_1 (10^{-7}dyn)$ & $\\kappa$ & $\\Theta$ & $\\zeta (\\mu m)$ & $k_BT (J)$ \\\\\n\\hline\n\\hline\n1{\\bf a-d} & $5$ & $$ & $$ & $$ & $$ & $3.75\\times10^{-3}$ & $18$ & $0.5$ & $$ & $0$\\\\\n\\cline{11-11} \n\\ref{fig:3}      & $$  & $$ & $$ & $$ & $$ & $$             & $$   & $$    & $$ & $$\\\\\n\\cline{2-2} \\cline{7-9}\n\\ref{fig:2}{\\bf a-b},\\;\\ref{fig:4} & $1$ & $-8\\times10^{-3}$ & $-0.5$ & $2.67$ & $0$  & $0.05$ & $0$ & $0$ & $3.55$ & $5\\times10^{-6}$ \\\\\n\\cline{2-2} \\cline{7-9}\n$$     & $$    & $$       & $$     & $$     & $$   & $0.05$ & $0$  & $0$   & $$   & $$\\\\\n\\ref{fig:2}{\\bf c}      & $5$    & $$ & $$ & $$ & $$  & $6.8\\times10^{-3}$  & $9$  & $0.5$ & $$     & $$ \\\\\n$$      & $$    & $$       & $$     & $$     & $$   & $3.75\\times10^{-3}$ & $18$ & $0.5$ & $$ & $$\\\\\n\\hline\n\\hline\n\\ref{fig:1}{\\bf e-h},\\;\\ref{fig:S1} & $0.02$ & $-4.5$ & $-0.5$ & $2.67$ & $3.56$ & $8.1$  & $0$  & $0$   & $3.55$ & $0$\\\\\n{\\ref{fig:5},\\;\\ref{fig:6}}            & $$   & $$   & $$   & $$   & $$   & $$ & $$ & $$  & $$   & $8\\times10^{-3}$\\\\\n\n\\hline\n\\end{tabular*}\n\\caption{\\label{tbl:GLdGparam} Parameters values excercised to mimic uniaxial (upper row) and biaxial \n(lower row) thermotropic NLC. A rectangular simulation box of size $80^2\\times160{\\mu}m^3$ with grid \nspacing $\\Delta x = \\Delta y= \\Delta z= 1{\\mu}m$ and time step $\\Delta t=1{\\mu}s$ is considered. We \nuse material parameters for $5CB$ at $T=33.65^{\\circ}$C, $\\epsilon_0=1,\\epsilon_a=\\pm1, \n\\epsilon_s=0.74\\epsilon_a$\\cite{colhird} and use earlier excercised material parameters for biaxial \nmedia\\cite{gralondej,bhatam}. Using equation (\\ref{eq:3}), we estimate $E_F=2\\times10^{-3} V\/{\\mu}m$ \nfor $5CB$ and with $|E|=E_F\\times(10^{-2},10^{-1},1)$, nondimensional parameters are $\\epsilon_1=\n1.5\\times(10^{-4},10^{-3},10^{-2})$, $\\epsilon_2=1.8\\times(10^{-3},10^{-2},10^{-1})$. In biaxial \nmedia, we use $|E|=1.5 V\/{\\mu}m<E_F, \\epsilon_1=0.14, \\epsilon_2=3.35$. $t_{on}=1ms$ for both uniaxial \nand biaxial problem, while $t_{off}=5ms$ for uniaxial and $t_{off}=\\infty$ for biaxial problem. \nParameters are defined in Appendix (section \\ref{app:1}).}\n\\end{table*}\n\nBy investigating beyond the uniform field assumption\\cite{olavmool}, in this article, we have developed \na fluctuating electronematics method based on the thermal description of the GLdG theory, with the physical \ncontrol over the role of each forcing to accurately describe thermal and electrokinetic phenomena in three \ndimensional NLC media. Using simple analytical argument, we provide an understanding \nof the underlying mechanism responsible for the outcome in both uniaxial and biaxial media in the free \ndraining limit at moderate to small electric field intensity, where the advective flow of the anisotropic \nmedia can be neglected. The interdependency between the topology in the orientational order of the \nmedia and morphology of the external field is elucidated through the measurement of disclination kinetics \nand morphology. It turns out that small magnitude of nonuniform electric field can significantly dilate \nthe coarsening kinetics of disclination network and can eradicate certain topological class of disclinations \nin biaxial media.\n\\begin{figure*}\n\\centering\n\\includegraphics[width=1.0\\textwidth, height=0.4\\textwidth]{figure2.pdf}\n\\caption{\\label{fig:2}{\\bf a,} Increment in disclination density with fluctuation amplitude. Note the \nthree different slopes corresponding to initial diffusive regime $\\to$ Porod's law regime $\\to$ late stage \ndiffusive regime. {\\bf b,} Comparison of disclination kinetics in a thermally fluctuating media with its \nathermal variant that corresponds to ({\\bf a}). A portion of the three-dimensional volume is shown for \nclarity, that displays domain decomposition before nucleation of disclinations $\\to$ intercommuting \ndisclinations $\\to$ contractile disclination loops. {\\bf c,} Increased elastic anisotropy leads to \nincreased disclination density with a prolonged Porod's law regime. Note that both thermal fluctuations \nand elastic anisotropy aids in early nucleation of the isotropic domain, but prolongs the disclination \nkinetics. The rendered colours in field values are indicated in figure \\ref{fig:1} and the arrow denotes \nthe increment direction. Material (computation) parameters are tabulated in Table \\ref{tbl:GLdGparam}.} \n\\end{figure*}\n\n\\begin{figure*}\n\\centering\n\\includegraphics[width=1.0\\textwidth, height=0.75\\textwidth]{figure3.pdf}\n\\caption{\\label{fig:3}{\\bf a-c,} Nonuniformity of electric flux lines along with the uniaxial order \nis sketched on slice plane $L_z\/8$ for negative dielectric anisotropy constant material (for example, \nbutylmethoxybenzylidene ($MBBA$)) when the magnitude of the non-dimensionalized scalar potential is \ncomparable to the orientation tensor (see Supplementary Movie S3). Disclinations in the bulk volume \nare also displayed to guide the eye. Uniformity in flux lines is gradually reached as the field is \nincreased to the Fr\\'{e}edericksz threshold $E_F$. {\\bf d-f,} Comparatively similar electric response \nof a positive dielectric anisotropy constant material (for example, $5CB$). {\\bf g,} Evolution of \ndisclination density per unit area displaying an increased life-span of $\\pi$ solitons with non-Markovian \nresponse during field cessation, {\\bf h,} elastic response of the global uniaxial order and {\\bf i,} \n(free) energy with varying field strength after onset and cessation of an unidirectional electric \nfield. Equilibrium response is also sketched for comparison. Material (computation) parameters are \ntabulated in Table \\ref{tbl:GLdGparam}.} \n\\end{figure*}\n\n\\section{\\sc \\bf Results}\n\\label{sec:3}\n\\subsection{\\sc \\bf Thermoelectrokinetic effects of disclination network in a coarsening uniaxial NLC}\n\\label{subsec:1} \nHere we systematically probe on the role played by the elasticity of the medium and various external \nforcing ($k_BT, {\\bf E}$) on the kinetics and microstructure of the string disclination assembly.\n \n{\\sc \\bf Role of thermal fluctuations and anisotropic elasticity.}\nThe disclination kinetics is immensely influenced by external agents like thermal and electric forces, \nanisotropic elastic effects or shear. To illustrate the role of thermal fluctuations, we estimate the \ndecay of disclination density per unit area in the degenerate elastic constant approximation without \nan electric field ($\\kappa=\\Theta=|E|=0$ in equation (\\ref{eq:6})). Using surface triangulation \nmethod\\cite{hjedae}, we calculate the total surface area of the disclinations in the media. Figure \n\\ref{fig:2}: frame {\\bf a} plots the same for three different values of $k_BT$ including the athermal \nscenario and in frame {\\bf b}, we show a portion of the three-dimensional volume that distinguishes \nthe disclination kinetics between the athermal and thermal scenario. In the athermal scenario, three \ndifferent regime with marked exponents emerges out in the evolution process, that had been previously \nquantified as diffusive regime $\\to$ Porod's law regime $\\to$ diffusive regime\\cite{bhatam}. The early \ndiffusive regime corresponds to domain coarsening before nucleation of disclinations ($t=0.84ms$), while \nPorod's law scale designates the defect annihilation kinetics ($t=1.4ms$) and finally, the late stage \ndiffusion is attributed to contraction of isolated loops ($t=11ms$). Clearly, thermal fluctuations tend \nto increase the disclination surface density without affecting the scaling laws, that is important for \nmaterials ({\\it e.g.} PAA) having transition temperature way above the room temperature. Fluid viscosity \n$\\eta$ can be obtained from the Stokes-Einstein relation $k_BT\/K\\eta$ = constant. However, frame {\\bf c} \nshows that disclination density in the athermal media not only increases for an increase in $\\kappa$, but \nalso  stretches the Porod's law regime. The slope changes from $0.8$ to $1$ as elastic anisotropy is \nincreased. While the slope of unity is also obtained in experiments with $5CB$\\cite{chdutuyur,chtuyur}, \nwe re-establish our earlier claim\\cite{bhatnucl} about the crucial contribution of the anisotropic \nelasticity of the medium. Intuitively, anisotropic elastic constant results into asymmetric diffusion \nconstants in Cartesian directions and thus brings asymmetry in the speed of $\\pm1\/2$ integer point \ndefects\\cite{todeyeo,niskmus}. The loss of area in forming a contractile loop is greatly reduced for \nhigher anisotropy.\n\nBefore we embark on discussing coarsening in the presence of an electric field, we revisit the coarsening \nkinetics when no electric field is present. As seen in Supplementary Movie S1, disclinations with higher \nline tension and curvature are energetically disfavoured, culminating in stretched strings that form \ncontractile loops after intercommuting with neighbouring strings. By taking into account the effect \nof viscous drag and length decrement of a string in forming a loop at time $t$, the disclination \nsurface density $\\rho$ is found\\cite{chdutuyur} to scale as $\\rho\\propto t^{-1}$. An identical scaling \nlaw is obtained for a planar disclination when equating the rate of change of the line tension per unit \nvolume with the energy density loss rate\\cite{klelav}. Our method accurately reproduces the slope of \n$1\\pm10^{-3}$ within the errorbar shown in figure \\ref{fig:3} (`no field' curve in frame {\\bf g}) \nwith the material parameters of $5CB$. Thermal fluctuations and elastic anisotropy lead to an increment \nin the disclination surface density at a given time and the period of disclination annihilation kinetics \nis extended without affecting the physical laws. \n\n{\\sc \\bf Role of an electric field.}\nNext, we elaborate on the thermal and electrokinetic effects on coarsening in uniaxial NLC after the onset \nand cessation of an unidirectional voltage pulse. The electric field is applied at an instant $t_{on}$ when \nonly disclinations with dipolar charge $\\pm1\/2$ are present, and, the medium is free of $\\pm1$ dipolar strings \nor point charges. The switch-off time of the field is set at an instant $t_{off}$ when the electric flux \nlines within the medium do not  change substantially. For a shallow quench below the supercooling line, for \nboth signs of the dielectric anisotropy constant and below the Fr\\'{e}edericksz threshold of the electric \nfield, the orientational order is comparable in magnitude with the non-dimensionalized electric potential. \nThus within the medium, the flux lines are very much distorted as seen in figure \\ref{fig:3}. \nInstead of being aligned along the field direction, the isotropic cores of the strings displaying reduced \nuniaxial order are little deformed by the electric force\\cite{veinblsmlavnob} and have a little \ncontribution in distorting the flux lines. This is attributed to the weak coupling of scalar order to the \nelectric field, unlike the director that strongly couples to the electric field. In no switch-off scenario \n($t_{off}\\to\\infty$), the nonuniformity of the flux lines is retained in the electrically forced nematic \nphase devoid of disclinations around $t\\sim35ms$ (not shown). The flux lines in figure \\ref{fig:3} (frames \n{\\bf a-f}), however, tend to gain uniformity as the electric field is increased towards the Fr\\'{e}edericksz\nthreshold for both signs of the dielectric anisotropy constant. As observed in frame {\\bf g}, \nelectrically forced strings are  little thinner due to a reduction of the surface density, but they are \nlong lived due to a dilated kinetics. After the field is switched off, disclination \nkinetics and surface density do not immediately return to the zero-field behavior, but lag for an interval \nof $\\sim 10ms$. Thus, the material retains a memory of the field onset in the process of exhibiting an \nelastic response. The electric field agitates the isotropic background towards an uniaxial medium, thus \nincreasing the uniaxial order (frame {\\bf h}) and non-monotonically decreasing the total free \nenergy (frame {\\bf i}). Though we obtain Brochard-Leg\\'{e}r lines connecting $\\pi$ solitons under \nan intense field above the Fr\\'{e}edericksz threshold, the lack of backflow in our method cannot reproduce \na ceasing motility of disclinations\\cite{veinblsmlavnob}. Rather, they annihilate at a faster rate due to \nthe uncompensated electric drag force. Electrokinetic effects under intense forcing, such as electroconvection \nor rheochaos, can be quantitatively captured only if backflow is systematically included. Unlike in colloidal \nsuspensions\\cite{donobhgabe}, the question of the existence of correlations between fluctuations in orientation \nand velocity has to be answered from experiments\\cite{zarsen} before attempting a numerical study in which \nbackflow effects are included.\n\\begin{figure}[t]\n\\centering\\includegraphics[width=0.45\\textwidth, height=0.35\\textwidth]{figure4.pdf}\n\\caption{\\label{fig:4} Spatial extent of surfaces of isolated uniaxial planar disclinations obtained from \nthe portion of the ${\\it xy}$-slice plane of the three-dimensional volume in figure \\ref{fig:2} (frame {\\bf \nb}). Colourbars are indicated in figure \\ref{fig:1}. Note the reduced fluctuation amplitude of $S$ for \n$|E|\\neq0$ compared to no field defect core.}\n\\end{figure}\n\nWe qualitatively argue about the slowing down of disclination kinetics for a planar isolated disclination \nloop. An in-plane estimate is valid for $5CB$ while twist constant is much smaller than splay or bend \nconstants\\cite{bhatnucl}. Also deep within the uniaxial phase where the director is aligned uniformly, \nFrank constant $K$ can sufficiently define the medium elasticity\\cite{bhmeads}. In such situations, planar \ndisclination energy per unit length is $\\mathcal{F}_{discl}=\\int_{\\mathbb{S}} d^2 x[K(\\pmb\\partial f)^2 - \n\\epsilon_0\\epsilon_a E^2 sin^2f]\/2$ where $f=k\\phi+c$, with $0\\le\\phi\\le2\\pi,\\; k=\\pm1\/2$ being the \ntopological charge, $\\mathbb{S}$ the bounding plane and $c$ a constant, defines a planar defect \nconfiguration ${\\bf n}=[cosf,sinf]$. Performing the surface integral, the reduced disclination energy \nper unit length is\n\\begin{equation}\n\\label{eq:7}\n\\mathcal{F}_{discl}=\\pi Kk^2ln\\Big({\\xi \\over \\zeta}\\Big) - {\\pi\\epsilon_0\\epsilon_a \\over 4k} \nE^2(\\xi^2-\\zeta^2) + \\mathcal{F}_{discl}^{c}, \n\\end{equation}\nwhere $\\mathcal{F}_{discl}^{c}$ is the disclination core energy. Equating the elastic energy \n$\\mathcal{F}_{discl}\/\\xi$ per unit area with the drag force $-\\eta\\partial_t{\\xi}$ per unit length \nyields contributions from (i) the equilibrium kinetics $\\xi\\sim t^{-1\/2}$ and (ii) the electric field \neffect $\\xi\\sim e^{\\nu t}$ with $\\nu=\\pi\\epsilon_0\\epsilon_aE^2\/8k\\eta$. While $\\nu$ is independent \nof the sign of $k$ or $\\epsilon_a$, $\\nu>0$ implies of a temporal reduction of the loop extinction \nkinetics. Physically this can be interpreted as a reduction in speed of approach between $\\pm1\/2$-charged \ntopological dipole within a charge-neutral loop due to the external forcing. As observed in figure \n\\ref{fig:3} (frame {\\bf g}) and in supplementary figure S1, nonuniform electric field substantially \nprolong the kinetics when compared to the uniform field scenario (see Supplementary Movie S4). \n\nTo shed light on the effect of electric forces on the disclination core structure in thermal uniaxial \nmedia with $\\epsilon_a>0$, in figure \\ref{fig:4} we sketch the spatial variation of the surface of $S$ \naround a planar defect for different field intensity and compare with the equilibrium scenario. The \nfluctuation amplitude at $S_{ueq}$ is reduced due to the application of an electric field, resulting \nin a reduction of the disclination surface density (figure \\ref{fig:3}, frame {\\bf g}). However, we do \nnot find any significant distortion of the core for an increasing field strength which indicates that \nthe field, unlike the director orientation, cannot influence the sufficiently isotropic core other \nthan a complete melting of the disclination at $E\\gg E_F$.\n\n\\subsection{\\sc \\bf Thermoelectrokinetic effects in coarsening biaxial NLC.}\nNext, we examine the role played by the isotropic elasticity of the medium and various external forcing \n($k_BT,{\\bf E}$) on the kinetics and microstructure of the string disclination assembly of different \nhomotopy class. We do not find excitingly different outcome when investigating the role of anisotropic \nelasticity and, thus, here we restrict ourselves in reporting results in $\\kappa=0$ limit, in par with \nother investigations\\cite{gralondej,bhatam}. \n\\begin{figure}[t]\n\\centering\n\\includegraphics[width=0.5\\textwidth, height=0.4\\textwidth]{figure5.pdf}\n\\caption{\\label{fig:5} Evolution \nof disclination density of $C_y$ class (left panel) and $C_z$ class (right panel) with fluctuation amplitude \nfor biaxial nematic media is displayed. The direction of the arrow shows the increment of surface density with \nfluctuation amplitude. In the inset to the left panel, biaxial order $B_2$ is sketched in which nucleation, \nintercommutation, and extinction by ring formation of disclinations is portrayed for clarity. Slope and \nerrorbar are indicated within the graphics. Material (computation) parameters are tabulated in Table \\ref{tbl:GLdGparam}.} \n\\end{figure}\n\n{\\sc \\bf Role of thermal fluctuations.}\nWe estimate the consequence of thermal fluctuations on the biaxial disclinations of class \\{$C_y,C_z$\\}. \nIn figure \\ref{fig:5}, we plot the evolution of disclination density for different values of $k_BT$ and \ncompare with the athermal scenario. To remind, in the inset to the left panel, we portray the early, \nintermediate and late stage of the disclination kinetics. Similar to the uniaxial disclinations, here \nwe also find that thermal fluctuation increases the disclination density per unit area with a comparable \nslope. The increase in slope for an increase in $k_BT$ during early stage of the kinetics for $C_y$ class \nhints for a delayed emergence of Porod's regime, that is absent in $C_z$ class. More interestingly, we \nobserve an equivalence of the $C_z$ class of biaxial disclinations with that of the $\\pm1\/2$-integer \ndisclinations in uniaxial nematics, both (i) in morphology (figure \\ref{fig:1}: frame {\\bf c} and frame \n{\\bf g}) and (ii) kinetics, as observed in the identical slope of $0.8$ in the Porod's law scaling regime \n(figure \\ref{fig:5}: right panel and figure \\ref{fig:2}: frame {\\bf a}). Also, the mismatch of the slope \nbetween the $C_y$ and $C_z$ class of disclinations (figure \\ref{fig:5}) suggests of minor influence \nwithin each other in the course of annihilation.\n\\begin{figure*}\n\\centering\n\\includegraphics[width=1.0\\textwidth, height=0.75\\textwidth]{figure6.pdf}\n\\caption{\\label{fig:6}{\\bf a-c,} Evolution and selection of disclinations of homotopy class $C_z$ for a negative dielectric \nanisotropy constant material under the onset of an applied field $E=1.5V\/{\\mu}m$. The presence of $C_y$ \ndisclinations at the early stage severely distorts the electric flux lines within the mesogen which regain \nuniformity as these disclinations are expelled from the medium (see Supplementary Movie S5-S6). {\\bf d-f,} \nSimilar response as previous for a positive dielectric anisotropy constant material, except that $C_y$ \ndisclinations are selected and at an early stage, $C_z$ disclinations widely distort the electric flux \nlines which gain uniformity after their expulsion. {\\bf g,} Decay of disclination surface density to \nportray quantitatively the selection of disclination class and {\\bf h,} contributions from volume and \nsurface energy to the total (free) energy of the medium. Equilibrium response is also sketched for \ncomparison. Material (computation) parameters are tabulated in Table \\ref{tbl:GLdGparam}.}\n\\end{figure*}\n\n{\\sc \\bf Role of an electric field.}\nTo conclude, we examine the thermal and electrokinetic effects on the coarsening of biaxial NLC when the \nsample is rapidly cooled from a disordered phase in the presence of a steady voltage pulse. Figure \n\\ref{fig:6} plots the instantaneous snapshots as well as the kinetic evolution of the medium. Although \nthe topological structure and kinetic pathway of disclinations in biaxial NLC have been predicted \nfor long\\cite{zapgolgol,klelav,lucsluc}, experimental advance to stabilize disclinations by avoiding \ncrystallization continues to be the {\\it holy grail} of research on thermotropic biaxial \nmesogens\\cite{madinasam}. Similar to the dilated kinetics of electrically forced uniaxial disclinations \n(see figure \\ref{fig:3}), we find in figure \\ref{fig:6} (frame {\\bf g}) that the onset of an electric \nfield increases the lifetime of biaxial disclinations of homotopy class \\{${C_y,C_z}$\\} at the initial \nstage for both sign of dielectric anisotropy. For either class of disclinations, \n$\\nu(=\\pi\\epsilon_0\\epsilon_aE^2\/8k\\eta)>0$ qualitatively explains the slowing down. For both signs of \nthe dielectric anisotropy constant of the material, flux lines are massively distorted in the presence \nof disclinations (frames {\\bf a,d}). After an interval of $\\sim5ms$, a clear asymmetry between the \ndisclination kinetics of different topology becomes evident. As shown in frames {\\bf b-c,e-f,g}, this \nresults in long-lived disclinations of either class with uniform electric field lines. This is attributed \nto an increment (decrement) of the total free energy with a positive (negative) value of the dielectric \nanisotropy constant (frame {\\bf h}). Thus, the dielectric energy has a strong influence in selecting \ndisclinations of the desired class as the bulk and elastic energies increase negligibly from the no \nfield scenario. Physically, the acceleration (or retardation) in the loop extinction kinetics at the \nlate stage can be interpreted as effective acceleration (or retardation) in speed of approach between \n$\\pm1\/2$-charged topological dipole within a charge-neutral loop of different class due to the electric \nforce. Consistent evidence of class selection is also obtained, however on a much longer timescale, \nfor values of the electric field magnitude much smaller than the Fr\\'{e}edericksz threshold value. \nHowever, the decay kinetics of disclinations of a particular homotopy class is accelerated in the \npresence of an intense electric field due to the absence of backflow in our model to compensate the \nelectric drag. We expect new phenomena in experiments on thermotropic biaxial media under an intense \nelectric field, perhaps similar to the behavior of uniaxial disclinations under an intense electric \nfield\\cite{veinblsmlavnob}.\n\n\\section{\\sc \\bf Discussions}\n\\label{sec:4}\nElectrorheology of line defects, with\\cite{aratan,ravzum,limastca} or without\\cite{todeyeo} \nparticulate inclusion, under an intense electric field have established the coupling of orientation \ntensor with hydrodynamics and uniformity in the electric field, though the effect of nonuniformity in \nthe electric field\\cite{tofuaronu,foronu,ucaronu,cumecakon} and the effect of thermal fluctuations are \nless explored. The effect of hydrodynamics is assumed to be negligible under moderate to weak electric \nfield intensity\\cite{beredw}. We have examined the role of thermal fluctuations and nonuniformity \nof electric field in this limit and have shown that fluctuating electronematics is a robust tool \nto mimic laboratory experiments\\cite{chdutuyur,veinblsmlavnob,niskcorazumu} {\\it in silico} for \nanisotropic NLC in three dimensions. From the structure of the orientation tensor, we present a \nsimple way to identify and classify the line defects and to compute physical quantities from the \ngeometry of disclinations.\n\nWe have shown how the spatial uniformity in electric field is gained in approaching the Fr\\'{e}edericksz \nlimit. Apart from modifying the kinetic pathway of the coarsening of athermal disclination network, \nthe external stimuli in terms of temperature fluctuations and local electric field essentially probe \ntwo emergent length scales: (i) interfacial correlation length between isotropic and nematic phase \nand (ii) radius of curvature of disclination loop. Neglecting any local heating effects due to the \nvariation of temperature, any change in the correlation length is attributed to the disclination \ncore size as well its geometric position within the three-dimensional volume. Although the evolution \nis temporally dilated compared to the no-field scenario, the external field cannot sufficiently modify \nthe radius of curvature of isotropic disclinations - thus the lines are not stretched along the direction \nof the electric field, rather they retain their shape even when the director gets aligned along or \nperpendicular to the field direction depending on the sign of material's dielectric anisotropy. \nThe inhomogeneity of the nematic orientation is manifest in the inherent nonuniformity of the local \nelectric field - resulting in the highly nonuniform electric flux lines within the sample. The \nelectric field induces a memory to the material that exhibit an elastic response and also induces \na kinetic asymmetry within disclinations of the different class. On the other hand, increase in thermal \nfluctuations tends to increase the disclination surface density. \n\nThis complex interaction can be intelligently engineered to yield a fascinating outcome in a more \ncomplex scenario, for example, fractal nematic colloids\\cite{hajamoejmura}, metadevices\\cite{zhekiv} \nand photonic applications\\cite{obayya}. The electric field induced kinetic asymmetry leading to the \nclass selection of biaxial disclinations can develop into novel materials in topologically similar \nsystems. Other than NLC, the presented work has resemblance with line defects in passive\\cite{rututu,bigahumur,kleman} \nand active\\cite{bartolo,drbrjoad} soft matter including conducting microwires\\cite{ralstan} and \nself-assembled resonators\\cite{frashu}, and thus has the potential to bring exciting applications \nin diverse systems.\n\n\\section{\\sc \\bf Acknowledgements}\nWe thank S.Ramaswamy, and C.Dasgupta for a constructive criticism and careful reading a version of \nthe manuscript. Including them, fruitful discussions with P.B.S.Kumar, R.Adhikari, and N.V.Madhusudana is \ngratefully acknowledged. We are thankful for a partial supercomputing support from {\\it Thematic Unit of \nExcellence on Computational Materials Science} at Indian Institute of Science. This work is supported by \nthe DST-INSPIRE grant number DST\/04\/2014\/002537 of Govt. of India.\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
{"text":"\\section{Introduction and results}\nThe duality between string theory on asymptotically AdS spaces and conformal gauge theories, usually known as the AdS\/CFT correspondence, has experienced an important evolution in the last few years. General non-BPS observables, as anomalous dimensions of composite operators and scattering amplitudes, can now be studied at high precision level providing sophisticated tests of the correspondence and sometimes offering non-trivial interpolating functions between weak and strong coupling. In both cases, the underlying integrability properties of the planar theory play a crucial role in the exact quantum evaluation and allow to follow the transition between the opposite regimes. Dramatic progresses have also concerned more traditional investigations, as the study of protected sectors of supersymmetric gauge theories: the introduction of powerful localization techniques makes now possible the exact computation of complicated path-integrals, providing again examples of interpolation between perturbative and asymptotic behaviors. It is tempting to speculate if the two different approaches, integrability and localization, could be somehow connected, at least in the computation of specific observables. \n\n\\noindent\nThis evocative possibility has been vigorously advocated in \\cite{Drukker:2011za} for a general class of Wilson loops in $\\mathcal{N}=4$ super Yang-Mills theory and concretely realized in a series of recent papers \\cite{Correa:2012at,Correa:2012nk,Drukker:2012de,Correa:2012hh}, where a new set of integral equations of TBA type, describing exactly a generalized cusp anomalous dimension, have been derived and checked against localization and perturbation theory at three loops. The result is striking and it contains, in principle, the all-order expression of the static potential between two heavy charged particles in four-dimensional maximally supersymmetric gauge theory.  \n\n\\noindent\nWilson loops are important observables in nonabelian gauge theories: they compute the potential between heavy colored probes, representing an order parameter for confinement \\cite{Wilson:1974sk} and encode a large part of information of the high-energy scattering between charged particles\n \\cite{Korchemsky:1988si,Korchemsky:1992xv}. In ${\\cal N}=4$ super Yang-Mills theory they play a prominent role, being the observables directly related to the fundamental string of the dual theory in $AdS_5 \\times S^5$ \\cite{Rey:1998ik,Maldacena:1998im}.  They are conjectured to calculate scattering amplitudes exactly \\cite{Bern:2005iz,Alday:2007hr} and in particular cases they are BPS operators \\cite{Dymarsky:2009si,Cardinali:2012sy}, whose quantum expectation value can be derived for any strength of the coupling constant \\cite{Erickson:2000af,Drukker:2000rr,Pestun:2007rz}. In \\cite{Drukker:2011za} a two-parameters family of Wilson loops in ${\\cal N}=4$ SYM has been proposed and studied both at weak and strong coupling: it consists of two rays in $\\mathbb{R}^4$ meeting at a point, with a cusp angle denoted by $\\pi-\\varphi$. Because the Maldacena-Wilson loops in ${\\cal N}=4$ SYM couple to a real scalar field, it is natural to consider different scalars on different rays, connected by an $R$-symmetry rotation of parameter $\\theta$. By varying continuously $\\varphi$ and $\\theta$ and performing suitable conformal transformations, these observables can be related to important physical quantities and to interesting BPS configurations. Mapping the theory to $S^3\\times\\mathbb{R}$ one obtains a pair of antiparallel lines, separated by angle $\\pi-\\varphi$ on the sphere, and derives the potential $V(\\lambda,\\varphi,\\theta)$ between two heavy $W$-bosons propagating over a large time $T$:\n\\begin{equation}\n\\langle \\mathcal{W}_{\\rm lines}\\rangle\\simeq \\exp \\left(-TV(\\lambda,\\varphi,\\theta)\\right).\n\\end{equation}\nThe usual potential in the flat space is easily recovered by taking the residue of $V(\\lambda, \\varphi ,0)$ as $\\varphi\\to\\pi$ \\cite{Drukker:2011za}. In the cusped version, the Wilson loop has the leading form\n\\begin{equation}\n\\langle\n\\mathcal{W}_{\\rm cusp}\n\\rangle\\simeq \\exp\\left(-\\Gamma_{cusp}(\\lambda, \\varphi,\\theta)\\log\\left(\\frac{\\Lambda}{\\epsilon}\\right)\\right),\\end{equation}\nand it turns out that $\\Gamma_{cusp}(\\lambda, \\varphi,\\theta)=V(\\lambda, \\varphi,\\theta)$, with $\\Lambda$ and $\\epsilon$ being IR and UV cut-offs respectively \\cite{Polyakov:1980ca,Brandt:1981kf}.  The cusped Wilson loops are strictly related to scattering amplitudes in Minkowski space: taking $\\varphi$ imaginary and large, $\\varphi=i\\phi$, produces to the so-called universal cusp anomalous dimension $\\gamma_{cusp}(\\lambda)$ \\cite{Korchemsky:1987wg}\n\\begin{equation} \\lim_{\\phi\\to\\infty}\\Gamma_{cusp}(\\lambda, i\\varphi,\\theta)=\\frac{\\varphi\\,\\gamma_{cusp}(\\lambda)}{4}.\\end{equation}\nRemarkably BPS configurations are also included in the family: for $\\theta=\\pm\\varphi$ the cusped Wilson loop is of Zarembo's type \\cite{ Zarembo:2002an}, implying the vanishing of  $\\Gamma_{cusp}(\\lambda, \\varphi,\\theta)$ in this case. By mapping conformally the rays into $S^2$ we recover instead the DGRT wedge, a well studied 1\/4 BPS operator \\cite{Drukker:2007qr,Bassetto:2008yf,Young:2008ed} belonging to the general class of loops on $S^3$ introduced in \\cite{Drukker:2007qr}. The quantum expectation value of the wedge is computed exactly by pertturbative two-dimensional Yang-Mills theory \\cite{Bassetto:1998sr}, a property shared with other DGRT loops on $S^2$  \\cite{Pestun:2009nn} and with a certain class of loop correlators \\cite{Bassetto:2009rt,Bassetto:2009ms,Giombi:2009ds,Giombi:2012ep}. We remark that path-integral localization properties are essential in order to derive these results. \n\n\\noindent\nIn the limit of small $\\varphi$, $\\Gamma_{cusp}(\\lambda, \\varphi,0)\\simeq-B(\\lambda)\\varphi^2$ computes also the radiation of a particle moving along an arbitrary smooth path \\cite{Correa:2012at} and an exact expression can be obtained by exploiting the BPS properties at $\\varphi=0$. A simple modification applies as well in expanding around the general BPS points $\\theta=\\pm\\varphi$, thanks to the knowledge of the all-orders expression of the DGRT wedge. Further, in \\cite{Drukker:2012de,Correa:2012hh}, a powerful set of TBA type integral equations for the generalized cusp was derived, using integrability: the explicit one-loop perturbative result \\cite{Drukker:2012de} and the three-loop expression of the near BPS limit \\cite{Correa:2012nk} were recovered as a check of the procedure.\n\n\\noindent\nIn view of these recent and exciting developments, it appears natural to wonder wether similar results could also be obtained for other superconformal gauge theories with integrable structures: the obvious choice is to investigate what happens in ${\\cal N}=6$ super Chern-Simons theories with matter, also known as ABJ(M) \\cite{Aharony:2008ug,Aharony:2008gk}. Wilson loops in ABJ(M) theory are still a rather unexplored subject: equivalence with scattering amplitudes has been shown at the second order in perturbation theory \\cite{Henn:2010ps,Chen:2011vv,Bianchi:2011dg,Bianchi:2011fc} and a quite mysterious functional similarity with their four-dimensional cousins seems to emerge. On the other hand, supersymmetric configurations, supported on straight lines and circles, have been discovered \\cite{Drukker:2009hy,Drukker:2008zx,Chen:2008bp,Rey:2008bh} and a localization formula, reducing the computation to an explicit matrix model average, has been derived \\cite{Kapustin:2009kz}. Concrete results at weak and strong coupling, using matrix model and topological string techniques, are presented in a beautiful series of papers \\cite{Marino:2009jd,Drukker:2010nc,Drukker:2011zy,Marino:2011eh}, where also various aspects of the partition function on $S^3$ have been discussed. Nevertheless many issues should be understood in order to extend the generalized cusp program. First of all it does not even exist a computation of the standard quark-antiquark potential nor of the conventional cusped Wilson loop. Secondly, BPS lines and circles appear in two fashions, distinguished by the degree of preserved supersymmetry: we have the 1\/6 BPS operators, originally defined in \\cite{Drukker:2008zx,Chen:2008bp} (see also \\cite{Rey:2008bh}), that are a straightforward extension of the Maldacena-Wilson loop to the three-dimensional case. The (quadratic) coupling with the scalar fields of the theory is governed by a mass matrix $M^I_J$, preserving an $SU(2)\\times SU(2)$ of the original $R$-symmetry, while gauge fields appear in the usual way. Although its simplicity, this kind of loop cannot be considered the dual of the fundamental string living in $AdS_4\\times \\mathbb{CP}^3$, because supersymmetries do not match \\cite{Drukker:2008zx}. The field theoretical partner of the fundamental string is instead the 1\/2 BPS loop discovered in \\cite{Drukker:2009hy} (see \\cite{Lee:2010hk} for a derivation arising from the low-energy dynamics of heavy $W$-bosons): the loop couples, in addition to the gauge and scalar fields of the theory, also to the fermions in the bi-fundamental representation of the $U (N )\\times U (M )$ gauge group. These ingredients are combined into a superconnection whose holonomy gives the Wilson loop, which can be defined for any representation of the supergroup $U(N |M )$. Supersymmetry is realized through a highly sophisticated mechanism, as a super-gauge transformation, requiring therefore the full non-linear structure of the path-ordering.  Actually both loops turn out to be in the same cohomology class, differing by a BRST exact term with respect the localization complex: their quantum expectation value should be therefore the same \\cite{Drukker:2009hy}. The above equivalence has not been checked at weak-coupling, where perturbative computations have been performed just for the 1\/6 BPS circle \n\\cite{Drukker:2008zx,Chen:2008bp,Rey:2008bh}, the presence of fermions complicates the calculations and rises delicate issues on the regularization procedure. Crucially there are also no examples of loops with fewer supersymmetries, including the known BPS lines and circle as particular cases: it would be interesting to find configurations of this type that could also help to understand better the mysterious cohomological equivalence. \n\n\\noindent\nThis paper represents a first step towards a systematic study of generalized cusps in ABJ(M) theories: similar configurations have been discussed, at strong coupling, in \\cite{Forini:2012bb}. We hope that our investigations could stimulate the application of integrability and cohomological techniques in the exact evaluation of non-BPS observables, such as the heavy-bosons static potential. Our main concern here is the construction of a generalized cusp using two 1\/2 BPS rays, the study of its supersymmetric properties and its quantum behavior at weak-coupling. The additional $R$-symmetry deformation is obtained by preserving different $SU(3)$ subgroups on the two lines: from the bosonic point of view this amounts to deform the mass-matrix $M^I_J$, by rotating two directions of opposite eigenvalues. The fermionic couplings experience a similar deformation and are also explicitly affected by the geometric parameter $\\varphi$, because they transform as spinors under spatial rotations. We study the supersymmetry shared by the two rays and we discover that for $\\theta=\\pm\\varphi$ two charges are still globally preserved: in this case the super-gauge transformations, encoding the supersymmetry variation on the two edges of the cusp, become smoothly connected at the meeting point.  A key observation made in the original paper of Drukker and Trancanelli \\cite{Drukker:2009hy} was that, for the 1\/2 BPS circle, only the trace of the super-holonomy turns out to be supersymmetric invariant and not its super-trace. Conversely the fermionic couplings were assumed to be anti-periodic on the loop: here we examine the same problem on the new BPS configurations. By performing an explicit conformal transformation, we map our cusp on $S^2$, obtaining a wedge: the fermionic couplings, constant on the plane, become space-dependent as an effect of the conformal map, and connected by a non-trivial rotation on the upper point of the wedge. In the BPS case this matrix simply appears as an anti-periodicity,  and therefore it is again the trace that leads to supersymmetric invariance. The loops constructed in this way are a sort of ABJ(M) version of the DGRT wedge \\cite{Drukker:2007qr} and preserve 1\/6 of the original supersymmetries. We consistently define our generalized cusp as the trace of the super-holonomy and attempt the computation of its quantum expectation value in perturbation theory. We observe two basic differences with the analogous four-dimensional computation performed in \\cite{Drukker:2011za}: first of all the effective propagators here, attaching on one side of the cusp only, are not automatically vanishing, as it happens for ${\\cal N}=4$ SYM in Feynman gauge, and the fermionic sector gives a divergent contribution at one-loop, that has to be regularized and renormalized. Secondly, because of the presence of a supergroup structure, involving fermions coupled to the external lines, it is not obvious to extend the non-abelian exponentiation theorem to this setting: we could not rely on such powerful device to reduce the amount of computations and to properly isolate the cusp anomalous dimensions. Concerning this second point we make the natural assumption that our cusped loops undergo through a ``double-exponentiation\" \n\\begin{equation}\n\\label{doppia}\n\\mathcal{W}=\\frac{M \\exp(V_{N})+ N \\exp(V_{M})}{N+M}, \n\\end{equation}\nwhere the generalized potentials\\footnote{With an abuse of language we have  referred  to $V_{N}$ and $V_{M}$ as the {\\it  generalised potentials.} Actually only  the coefficient of the $1\/\\epsilon$ pole  has this meaning.}\n $V_N$ and $V_M$ are simply related by exchanging $N$ with $M$. A highly non-trivial check of the above assumption is the actual exponentiation of the one-loop term, constraining in particular the structure of the double-poles (in the dimensional regularization parameter $\\epsilon=(3-D)\/2$) appearing at two-loops. We find a perfect agreement of our results with the double-exponentiation hypothesis, recovering at the second order in perturbation theory the quadratic contributions coming from the first order one. Our final expression for the unrenormalized $V_N$ is\n\\begin{align}\nV_{N}=&\\left(\\frac{2\\pi}{\\kappa}\\right) N\\left(\\frac{ \\Gamma(\\frac{1}{2}-\\epsilon)}{4\\pi^{3\/2-\\epsilon}}\\right)\n(\\mu L)^{2\\epsilon}\\left[\\frac{1}{\\epsilon}\\left(\\frac{\\cos\\frac{\\theta}{2}}{\\cos\\frac{\\varphi}{2}}-2\\right)-2\\frac{\\cos\\frac{\\theta}{2}}{\\cos\\frac{\\varphi}{2}}\\log \\left(\\sec \\left(\\frac{\\varphi }{2}\\right)+1\\right)\\right]+\\nonumber\\\\\n&\\left(\\frac{2\\pi}{\\kappa}\\right)^{\\!\\!2}N^2\\!\\left(\\frac{\\Gamma\\!\\left(\\frac{1}{2}-\\epsilon\\right)}{4 \\pi^{3\/2-\\epsilon}}\\right)^2(\\mu L)^{4\\epsilon}\\left[\\frac{1}{\\epsilon}\\log\\left(\\cos\\frac{\\varphi}{2}\\right)^2\\left(\\frac{\\cos\\frac{\\theta}{2}}{\\cos\\frac{\\varphi}{2}}-1\\right)+O(1)\\right].\n\\end{align}\nHere $k$ is the Chern-Simons level, $L$ is the length of the lines and $\\mu$ is the renormalization scale introduced by dimensional regularization. To extract the cusp anomalous dimension we have to carefully subtract the divergences coming from single-leg diagrams: for closed contours in four dimensions these are usually associated, in a generic gauge theory, to a linear divergence proportional to the perimeter loop \\cite{Polyakov:1980ca}. In the smooth case once subtracted this perimeter term. the standard lagrangian renormalization makes the quantum expectation value finite \\cite{Brandt:1981kf,Exp}. \n\n\\noindent\nWhen open contours are considered the situation changes and some subtleties in the renormalization procedure arise: a systematic analysis of these problems have been performed in eighties \\cite{Aoyama:1981ev,Dorn1,Dorn2,Knauss:1984rx,Dorn:1986dt} and (somehow) forgotten. The outcome is essentially contained in the introduction of a further a gauge-dependent renormalization constant, sometimes called $Z_{\\rm open}$, taking into account shape-independent extra-divergencies associated to the end-points of the contour. To isolate the true gauge-invariant cusp-divergence these spurious contributions should be subtracted because, in general, they appear for finite lenght of the lines. ${\\cal N}=4$ SYM in Feynman gauge represents a lucky situation in which, due to the peculiar combination of the gauge\/scalar propagator, these additional effects are not present. We remark that in general $\\alpha$-gauge a $Z_{\\rm open}$ should be taken into account. We will carefully review all these topics in subsec. \\ref{outcome}. \n\\noindent\n  \nIn three dimensions the superconformal case, due to the fermionic couplings, inevitably implies the appearence of the spurios single-length contributions: we will carefully discuss the subtraction procedure, examing in details the paradigmatic case of the 1\/2 BPS infinite-line, and we hope to clarify the structure of the divergences for these family of loops. We will also comment on the difference between the 1\/2 BPS and 1\/6 BPS cases, showing that at finite-length the cohomological equivalence is broken by boundary terms, generating the unexpected divergence at quantum level. \n\nOur final receipt amounts to subtract, in the second order computation,  the one-loop poles associated to single line diagrams, normalizing in this way the final result to the straight line, 1\/2 BPS contour\n\\begin{align}  \n\\!\\!V^{\\rm Ren.}_{N}=&\\left(\\frac{2\\pi}{\\kappa}\\right) N\\left(\\frac{ \\Gamma(\\frac{1}{2}-\\epsilon)}{4\\pi^{3\/2-\\epsilon}}\\right)\\!\n(\\mu L)^{2\\epsilon}\n\\!\\!\\left[\\frac{1}{\\epsilon}\\left(\\frac{\\cos\\frac{\\theta}{2}}{\\cos\\frac{\\varphi}{2}}-1\\right)\\!\\!-2\\frac{\\cos\\frac{\\theta}{2}}{\\cos\\frac{\\varphi}{2}}\\log \\left(\\sec \\left(\\frac{\\varphi }{2}\\right)\\!+\\!1\\right)\\!\\!+\\log 4\\right]\\!+\\nonumber\\\\\n&+\\left(\\frac{2\\pi}{\\kappa}\\right)^{\\!\\!2}N^2\\!\\left(\\frac{\\Gamma\\!\\left(\\frac{1}{2}-\\epsilon\\right)}{4 \\pi^{3\/2-\\epsilon}}\\right)^2(\\mu L)^{4\\epsilon}\\left[\\frac{1}{\\epsilon}\\log\\left(\\cos\\frac{\\varphi}{2}\\right)^2\\left(\\frac{\\cos\\frac{\\theta}{2}}{\\cos\\frac{\\varphi}{2}}-1\\right)+O(1)\\right].\n\\end{align}\nFrom the above expression we can easily recover the quark-antiquark potential\\footnote{Actually we have two potentials, $V^{(s)}_N$ and $V^{(s)}_M$, associated respectively to singlets in the $N\\times\\bar{N}$ and $M\\times\\bar{M}$ direct product.}, taking the limit $\\varphi\\to \\pi$ and following the prescription of \\cite{Drukker:2011za}\n\\begin{equation}\nV^{(s)}_{N}(R)=\\frac{N}{k}\\frac{1}{R}-\\left(\\frac{N}{k}\\right)^2\\frac{1}{R}\\log\\left(\\frac{T}{R}\\right).\n\\end{equation}\n\nWe find a logarithmic, non-analytic term in $T\/R$ at the second non-trivial order that, as in four dimensions, is expected to disappear when resummation of the perturbative series is performed. In the opposite limit, for large imaginary $\\varphi$, we get the universal cusp anomaly (using the four-dimensional definition)\n\\begin{equation}\n\\gamma_{cusp}=\\frac{N^2}{k^2},\n\\end{equation}\nreproducing the result obtained directly from the light-like cusp \\cite{Henn:2010ps}.\n\n\\noindent\nThe plan of the paper is the following: in Section 2 we review the construction of 1\/2 BPS Wilson lines in ABJ models, giving us the possibility to introduce the peculiar structures entailed by maximal supersymmetric loops in ${\\cal N}=6$ super Chern-Simons-matter theories. Section 3 is devoted to the explicit  realization of the generalized cusp: we obtain the appropriate bosonic and fermionic couplings and their deformations and discuss how the supersymmetry properties depends on the relevant parameters. The conformal transformation, mapping the cusp on a wedge of $S^2$, is also presented: the periodicity properties of the fermions are derived and BPS observables are obtained taking the trace of the super-holonomy. In Section 4 we start the quantum investigation computing the expectation value at the first order in perturbation theory. The two-loop analysis is contained in Section 5. The final result, obtained by summing up all the contributions and performing the renormalization procedure is presented in Section 6, where the peculiar divergences structure of these observables is carefully discussed. We present a rather detailed review of known facts on the renormalization of closed, open and cusped Wilson loops, that we think will clarify the apparent intricacy of our subtraction procedure and unveil its gauge-independent meaning. Some conclusions and outlooks appear in Section 7. We complete the paper with some appendices, containing our conventions and the technical details of the computations.\n\n\\section{1\/2 BPS straight-line in ABJ theories}\n\\label{sec2}\nWe start by reviewing the construction of the $1\/2-$BPS Wilson line  given in \\cite{Drukker:2009hy,Lee:2010hk}: the mechanism leading to its gauge  invariance is carefully reconsidered, since it is substantially different from the four dimensional analogue.  \n\n\nThe  central idea  of \\cite{Drukker:2009hy} is to replace the obvious $U(N)\\times U(M)$  gauge  connection with  a super-connection\\begin{equation}\n\\label{superconnection}\n \\mathcal{L}(\\tau) \\equiv -i \\begin{pmatrix}\ni\\mathcal{A}\n&\\sqrt{\\frac{2\\pi}{k}}  |\\dot x | \\eta_{I}\\bar\\psi^{I}\\\\\n\\sqrt{\\frac{2\\pi}{k}}   |\\dot x | \\psi_{I}\\bar{\\eta}^{I} &\ni\\hat{\\mathcal{A}}\n\\end{pmatrix} \\ \\  \\ \\ \\mathrm{with}\\ \\ \\ \\  \\left\\{\\begin{matrix} \\mathcal{A}\\equiv A_{\\mu} \\dot x^{\\mu}-\\frac{2 \\pi i}{k} |\\dot x| M_{J}^{\\ \\ I} C_{I}\\bar C^{J}\\\\\n\\\\\n\\hat{\\mathcal{A}}\\equiv\\hat  A_{\\mu} \\dot x^{\\mu}-\\frac{2 \\pi i}{k} |\\dot x| \\hat M_{J}^{\\ \\ I} \\bar C^{J} C_{I},\n\\end{matrix}\\ \\right.\n\\end{equation}\nbelonging to the super-algebra\\footnote{In Minkowski space-time, where $\\psi$ and $\\bar\\psi$ are related by complex conjugation, $\\mathcal{L}(\\tau)$ belongs to $\\mathfrak{u}(N|M)$ if $\\bar\\eta=i (\\eta)^{\\dagger}$. In Euclidean space, where the reality condition among spinors are lost,  we shall deal with the complexification  of this group $\\mathfrak{sl}(N|M)$.} of $U(N|M)$. In \\eqref{superconnection} the coordinates $x^{\\mu}(\\tau)$  \ndraw the contour along which the loop operator is defined,  while  $M_{J}^{\\ \\ I}$, $\\hat M_{J}^{\\ \\ I}$, $\\eta_{I}^{\\alpha}$ and $\\bar{\\eta}^{I}_{\\alpha}$ are  free parameters. The latter two, in particular, are Grassmann even quantities even though they transform in the spinor representation of the Lorentz group.\n\nThe dependence of $\\mathcal{L}(\\tau)$ on the fields is largely dictated by dimensional analysis and transformation properties. Since the {\\it classical} dimension of the  scalars in $D=3$ is $1\/2$,  they could only appear as bilinears,  transforming  in the adjoint and thus entering in the diagonal blocks together with the gauge fields. Instead  the fermions  have  dimension $1$ and should appear linearly. Since they transform in the bi-fundamentals of the gauge group, they are naturally placed in the off-diagonal entries of the matrix \\eqref{superconnection}. \n\nWhen the contour $x^{\\mu}(\\tau)$ is a straight-line $S$, the  invariance  under translations along the direction  defined by $S$ ensures that all the couplings can be chosen to be  independent of $\\tau$, {\\it i.e.} constant. \nMoreover the requirement of having  an unbroken  $SU(3)$ $R-$symmetry, as that of the dual string configuration, restricts  the couplings $M_{J}^{\\ \\ I}$,\\  $\\hat M_{J}^{\\ \\ I}$,\\ $\\eta_{I}^{\\alpha}$\\  and\\  $\\bar{\\eta}^{I}_{\\alpha}$ to be   of the \nform\n\\begin{equation}\n\\begin{split}\n\\label{cc}\n\\eta_{I}^{\\alpha}=n_{I} \\eta^{\\alpha},\\ \\ \\ \\   \\bar\\eta^{I}_{\\alpha}=\\bar n^{I} \\bar\\eta_{\\alpha},\\ \\ \\ \\ \nM_{J}^{\\ \\ I}=p_{1}\\delta^{I}_{ J}-2 p_{2} n_{J}  \\bar n^{I},\\ \\ \\ \\ \n\\widehat M_{J}^{\\ \\ I}=q_{1} \\delta^{I}_{J}-2 q_{2} n_{J} \\bar n^{I}.\n\\end{split}\n\\end{equation}\nHere $n_{I}$  and $\\bar n^{I}$ are two complex conjugated vectors which transform in the fundamental \nand anti-fundamental representation and determine the embedding of \nthe $SU(3)$ subgroup in $SU(4)$\\footnote{In the internal $R-$symmetry space $n_{I}$ identifies the direction preserved by the action of the  $SU(3)$ subgroup}. By rescaling $\\eta^{\\alpha}$ and $\\bar\\eta_{\\alpha}$, we  can always choose $n_{I}\\bar n^{I}=1$. The parameters $p_{i}$ and $q_{i}$  in the definition of $M$ and $\\hat M$ instead  control the eigenvalues of the two matrices.\n\nThe  free  parameters appearing in \\eqref{cc} can be then  constrained by imposing that the resulting Wilson loop is globally supersymmetric. This issue is subtle: the usual requirement $\\delta_{\\rm susy}\\mathcal{L}(\\tau)=0$ does not  yield  any 1\/2 BPS solution indeed. We  just obtain   loop operators which are merely  bosonic ($\\eta=\\bar\\eta=0$) and   at most $1\/6$ BPS \\cite{Drukker:2008zx,Rey:2008bh}. In order to obtain 1\/2 BPS solution, we must replace  $\\delta_{\\rm susy}\\mathcal{L}(\\tau)=0$  with  the weaker condition \\cite{Drukker:2009hy,Lee:2010hk}\n\\begin{equation}\n\\label{var1}\n\\delta_{\\rm susy}\\mathcal{L}(\\tau)=\\mathfrak{D}_{\\tau} G\\equiv\\partial_{\\tau} G+ i\\{ \\mathcal{L},G],\n\\end{equation}\nwhere  the r.h.s. is the super-covariant derivative  constructed out of the connection  \n$\\mathcal{L}(\\tau)$  acting on a super-matrix $G$ in $\\mathfrak{u}(N|M)$. The requirement \\eqref{var1} \nassures that the action of the relevant  supersymmetry charges translates into an infinitesimal\n super-{\\it gauge} transformation for $\\mathcal{L}(\\tau)$ and thus   the ``{\\it traced}'' loop operator is invariant. \n\nNow we shall recapitulate  the analysis  of \\cite{Drukker:2009hy} leading to fix the  free parameters in \\eqref{cc}.\nHowever, for future convenience, we shall present the result  in a  {\\it covariant} notation {\\it i.e.} without referring to a specific form of the straight line.\n\nWe start by considering the structure of the infinitesimal gauge parameter in \\eqref{var1}.\nSince the supersymmetry transformation of the bosonic fields does not contain any derivative of the fields,  the super-matrix $G$ in \\eqref{var1}\nmust be anti-diagonal \n\\begin{equation}\n\\label{susy2}\nG=\\begin{pmatrix}\n0 & g_{1}\\\\\n\\bar g_{2}  & 0\n\\end{pmatrix}\\ \\ \\Rightarrow \\  \\  \\mathfrak{D}_{\\tau} G=\\begin{pmatrix}\n\\sqrt{\\frac{2\\pi}{k}}  |\\dot x | (\\eta_{I}\\bar\\psi^{I} \\bar g_{2}+g_{1}\\psi_{I}\\bar \\eta^{I}) &\\mathcal {D}_{\\tau} g_{1}\\\\\n\\mathcal{D}_{\\tau }\\bar g_{2}  & \\sqrt{\\frac{2\\pi}{k}}  |\\dot x | (\\bar g_{2}\\eta_{I}\\bar\\psi^{I} +\\psi_{I}\\bar \\eta^{I} g_{1}) \n\\end{pmatrix} .\n\\end{equation}\nHere $\\mathcal{D}_{\\tau}$ is the covariant derivative constructed out of  the {\\it dressed} bosonic connections $\\mathcal{A}$ and $\\hat{\\mathcal{A}}$ and given by\n\\begin{equation}\n\\begin{aligned}\n\\mathcal{D}_\\tau g_{1}&= \\partial_\\tau  g_{1} + i (\\mathcal{A}\\,  g_{1} - g_{1}\\, \\hat {\\mathcal{A}})\\,, \\ \\ \\ \\ \n\\mathcal{D}_\\tau \\bar g_{2} &= \\partial_\\tau \\bar g_{2} \n- i (\\bar g_{2}\\,\\mathcal{A}-   \\hat{\\mathcal{A}}\\, {\\bar g}_{2}) .\\,\n\\end{aligned}\n\\end{equation}\n The condition \\eqref{var1} for the anti-diagonal entries  first constrains   the form \n of the spinor $\\eta$ and $\\bar\\eta$  to obey the two conditions\n\\begin{equation}\n\\label{orto1}\n{(\\dot{x}^{\\mu}\\gamma_{\\mu})_{\\alpha}^{\\ \\ \\beta}}=\\frac{1}{(\\eta\\bar\\eta)} |\\dot x|(\\eta^{\\beta} \\bar\\eta_{\\alpha}+\n\\eta_{\\alpha} \\bar\\eta^{\\beta})\\ \\ \\ \\ \\ \\   (\\eta^{\\beta} \\bar\\eta_{\\alpha}-\n\\eta_{\\alpha} \\bar\\eta^{\\beta})=(\\eta\\bar\\eta)\\delta^{\\beta}_{\\alpha},\n\\end{equation}\nwhich assure that the covariant derivatives appearing  in the supersymmetry transformations of  $\\psi\\bar\\eta$ and $\\eta\\bar\\psi$  are  only evaluated along the circuit.  The value of the parameters $p_{i}$ and $q_{i}$, appearing in the matrices $M$ and $\\hat M$, is equal to $1$ for the same reason. \n\nThe requirement \\eqref{var1} for the diagonal entries  does not yield, instead, new conditions, simply fixing the normalization \n\\begin{equation}\n\\label{norm}\n\\eta\\bar\\eta=2 i.\n\\end{equation}\nIn particular the vector $n_{I}$ continues to be unconstrained. \n\nThe origin of the superconnection was also investigated from the point of view of the low-energy dynamics of heavy {\\it ``W-bosons''}  in \\cite{Lee:2010hk}. It was shown that when the theory is higgsed preserving half of the total supersymmetry, the corresponding low-energy Lagrangian enjoys a larger gauge invariance, given by the supergroup $U(N|M)$. The light fermions do not decouple from the dynamics, at variance with the case of ${\\cal N}=4$ SYM, and their interactions with heavy $W$-bosons are described by $\\eta_{I}^{\\alpha},\\bar{\\eta}^{I}_{\\alpha}$. The role of the mass-matrix is instead played by $M_{J}^{\\ \\ I},\\widehat M_{J}^{\\ \\ I}$. This result unveils the physical nature of the potential related to the rectangular Wilson loops, constructed with 1\/2 BPS lines in ABJ(M) theories.   \n\n\nArmed with the explicit form for the couplings, we can find  twelve supercharges \\cite{Drukker:2009hy}   whose action on $\\mathcal{L}(\\tau)$ \ncan be cast into the form \\eqref{var1}. There are six supercharges of the Poincar\\`e type\\footnote{Recall that the counting is performed in terms of complex supercharges in Euclidean space-time,\nwhile we use {\\it real} supercharges in Minkowski signature.},\n \\begin{equation}\n \\label{susypar}\n \\bar\\theta^{IJ\\beta\n= (\\bar\\eta^{I\\beta} \\bar v^{J}-\\bar\\eta^{J\\beta} \\bar v^{I})\n-i\\epsilon^{IJKL} \\eta_{K}^{\\beta} u_{L}=\n(\\bar n^{I} \\bar v^{J}-\\bar n^{J} \\bar v^{I})\\bar\\eta^{\\beta}\n-i\\epsilon^{IJKL} n_{K} u_{L}\\eta^{\\beta},\n \\end{equation} \nparametrized by  two  vectors $u^{I}$ and $\\bar v^{I}$ that satisfy $(n_{I}\\bar v^{I})=(\\bar n^{I} u_{I})=0$.  We remark that these vectors  are really  independent in Euclidean signature, while  $\\bar{v}^{I}=u_{I}^{\\dagger}$ as a result of  the reality conditions  present in  the Minkowski case.\n Next to the above $\\bar\\theta^{IJ}$  we  can also identify six super-conformal   charges\\footnote{We have parametrized a generic supercharge as follows\n\\[\\bar\\Theta^{IJ}=\\bar\\theta^{IJ}+x^{\\mu}\\gamma_{\\mu}\\bar\\vartheta^{IJ},\n\\]\nwhere $\\bar\\theta^{IJ}$ generates the Poincar\\`e supersymmetries, while $\\bar\\vartheta^{IJ}$ yields the conformal ones.\n} $\\bar\\vartheta^{IJ\\beta}$, whose structure is again given by an expansion of  the form \\eqref{susypar}. \nThe origin of this second set of supersymmetries is easily understood: they are obtained by combining\nthe  Poincar\\`e supercharges \\eqref{susypar} with a special conformal transformation in the direction associated to the straight-line.\n\nThe  analysis presented in \\cite{Drukker:2009hy,Lee:2010hk} also provides the explicit form of the gauge function  \nin terms of the  scalar fields, the spinor couplings and the supersymmetry parameters $\\bar\\theta^{IJ}$.\nIn our notation,  they take the form \n\\begin{equation}\n\\label{gaugefunc}\ng_{1}=  2\\sqrt{\\frac{2\\pi}{k}}  n_{K}(\\eta\\bar\\theta^{KL})C_{L}\\ \\  \\ \\ \\ \\  \\mathrm{and} \\ \\ \\ \\ \\ \\ \\ \n\\bar g_{2}=-\\sqrt{\\frac{2\\pi}{k}}   \\varepsilon_{IJKL} \\bar n^{K}(\\bar\\theta^{IJ}\\bar\\eta) \\bar C^{L}.\n\\end{equation} \nNow we come back to  analyze the issue  of  supersymmetry invariance  for a generic  Wilson loop defined  by \\eqref{superconnection} when its variation can be cast into the form \\eqref{var1}.   The finite transformation  of the untraced operator \n\\begin{equation}\nW(\\tau_{1},\\tau_{0})=\\mathrm{P}\\exp\\left(-i \\int_{\\tau_{0}}^{\\tau_{1}} d\\tau \\mathcal{L}(\\tau)\\right),\n\\end{equation}\nunder the  gauge transformation  generated by $G(\\tau)$ in \\eqref{susy2}, can be written as \\cite{Lee:2010hk}\n\\begin{equation}\n W(\\tau_{1},\\tau_{0})\\mapsto  U^{-1} (\\tau_{1}) W(\\tau_{1},\\tau_{0}) U(\\tau_{0}),\n\\end{equation} \nwhere $U(\\tau)=e^{i G(\\tau)}$.  For a closed path $\\gamma$ $[\\gamma(\\tau_{0})=\\gamma(\\tau_{1})]$, \nwe must carefully consider the boundary conditions  obeyed by the gauge functions $g_{1}$ and \n$\\bar g_{2}$ in order to define the gauge invariant operator. If they are {\\it periodic}, {\\it i.e.} $g_{1}(\\tau_{0})=g_{1}(\\tau_{1})$ and $\\bar g_{2}(\\tau_{0})=\\bar g_{2}(\\tau_{1})$, we find that $U(\\tau_{0})=U(\\tau_{1})$ and a gauge invariant operator  is obtained  by taking\nthe usual super-trace\n\\begin{equation}\n\\mathcal{W}=\\mathrm{Str}(W).\n\\end{equation}\nActually it is the super-trace to be invariant under similitude transformations.  However we can have  different situations: in \\cite{Drukker:2009hy} it was examined another 1\/2-BPS loop, the circle, and pointed out that the function $g_{1}$ and $\\bar g_{2}$ are anti-periodic in this case. Consequently the untraced operator, because $U(\\tau_{1})=U^{-1}(\\tau_{0})$, transforms as follows\n\\begin{equation}\n W(\\tau_{1},\\tau_{0})\\mapsto  U (\\tau_{0}) W(\\tau_{1},\\tau_{0}) U(\\tau_{0}).\n\\end{equation}\nTo construct a supersymmetric operator, we first observe that\n\\begin{equation}\\mbox{\\scriptsize $\\begin{pmatrix} \\mathds{1} & 0\\\\ 0 &-\\mathds{1}\\end{pmatrix}$}U(\\tau_{0}) \\mbox{\\scriptsize$\\begin{pmatrix} \\mathds{1} & 0\\\\ 0 &-\\mathds{1}\\end{pmatrix}$}=U^{-1}(\\tau_{0})\n\\end{equation}\n  for a gauge transformation generated by the matrix $G$ in \\eqref{susy2}.\nThen   the operator \n\\begin{equation}\n\\mathcal{W}=\\mathrm{Str}\\left[\\mbox{\\footnotesize $\\begin{pmatrix} \\mathds{1} & 0\\\\ 0 &-\\mathds{1}\\end{pmatrix}$}\\mbox{$\\displaystyle\\mathrm{P}\\!\\exp\\left(-i \\int_{\\tau_{0}}^{\\tau_{1}} d\\tau \\mathcal{L}(\\tau)\\right)$}\\right]\\equiv\\mathrm{Tr}\\left[\\mbox{$\\displaystyle\\mathrm{P}\\!\\exp\\left(-i \\int_{\\tau_{0}}^{\\tau_{1}} d\\tau \\mathcal{L}(\\tau)\\right)$}\\right]\n\\end{equation}\nturns out to be  invariant.  In the case of a straight line the situation  is more intricate, since we deal with an open infinite circuit. The invariance under supersymmetry is recovered by choosing a set of suitable boundary conditions for the fields, in particular for the scalars appearing in the definition of $g_{1}$ and $\\bar \ng_{2}$. The naive statement that they must vanish when $\\tau=\\pm \\infty$ seems to leave open a double\npossibility for defining a supersymmetric operator\n\\begin{equation}\n\\label{Wpm}\n\\begin{split}\n\\mathcal{W}_{-}=&\\frac{1}{N-M}\n\\mathrm{Str}\\left[\\mbox{$\\displaystyle\\mathrm{P}\\!\\exp\\left(-i \\int_{\\tau_{0}}^{\\tau_{1}} d\\tau \\mathcal{L}(\\tau)\\right)$}\\right]\\   {\\mathrm{or}}\\\\\\mathcal{W_{+}}=&\\frac{1}{N+M}\n\\mathrm{Tr}\\left[\\mbox{$\\displaystyle\\mathrm{P}\\!\\exp\\left(-i \\int_{\\tau_{0}}^{\\tau_{1}} d\\tau \\mathcal{L}(\\tau)\\right)$}\\right],\n\\end{split}\n\\end{equation}\nsince $U(\\pm\\infty)=\\mathds{1}$ and $W(\\tau_{1},\\tau_{0})$ itself is invariant. We shall consider in our explicit quantum computation the second possibility: as we will see, the trace is the correct option to generate BPS observables from closed contours, connected to ours through conformal transformations. It also seems to provide a result consistent with the interpretation of the Wilson loop in terms of quark-antiquark potentials.\n\n\n\n\\section{The generalized cusp}\nWe discuss here in detail  the Wilson loop observables we will study in the rest of the paper. After constructing the bosonic and fermionic couplings for the generalized cusp, we study the possibility to find novel BPS configurations. We determine the BPS conditions for the cusp parameters and derive the explict form of the related supercharges. Finally we map our new configurations on the sphere $S^2$, by means of conformal transformations, and we obtain a non-trivial BPS deformation of the BPS circle constructed in \\cite{Drukker:2009hy}.\n\n\\subsection{Bosonic and fermionic couplings}\n\\label{subseccoupl}\nWe start by considering  the  theory on the Euclidean space-time.  We shall consider two rays  in the plane $(1,2)$ intersecting at the origin as  illustrated  in fig.  \\ref{cusp1}. The angle between the rays is $\\pi-\\varphi$, such that for $\\varphi= 0$ they form a continuous straight line.\nThe path in fig. \\ref{cusp1} is given by\n \\begin{figure}[ht]\n\\centering\n\t\\includegraphics[width=.6 \\textwidth]{cuspconventions.pdf}\n\\caption{\\label{cusp1} Eq. \\eqref{cusp} represents a planar cusp, whose angular extension is given by $\\pi-\\varphi$.}\n\\end{figure}\n\\begin{equation}\n\\label{cusp}\n x^{0}=0 \\ \\ \\ \\ \\ x^{1}=s\\cos\\frac{\\varphi}{2} \\ \\ \\ \\ \\  x^{2}=|s|\\sin\\frac{\\varphi}{2}\\ \\ \\ \\ \\  -\\infty\\le s\\le \\infty.\n\\end{equation}\n The fermionic couplings on each straight-line  possess the  factorized structure discussed in the previous section, {\\it i.e.}\n\\begin{equation}\n\\label{cuspcoupling}\n\\eta_{i M}^{\\alpha}=n_{i M} \\eta^{\\alpha}\\ \\ \\ \\ \\ \\mathrm{and}\\ \\ \\ \\ \\  \\bar\\eta^{M}_{i\\alpha}=\\bar n_{i}^{M} \\bar\\eta_{i\\alpha}.\n\\end{equation}\nThe  additional  index $i=1,2$ in \\eqref{cuspcoupling} specifies which edge of the cusp  we are considering.\nFor $i=\\mathrm{1}$,  $\\eta_{1\\alpha}$ is constructed as the  eigenspinor of eigenvalue $1$ of the matrix \n$\\frac{\\dot{x}_{1}^{\\mu}}{|\\dot x_{1}|}\\gamma_{\\mu}$:\n\\begin{equation}\n\\frac{\\dot{x}_{1}^{\\mu}}{|\\dot x_{1}|}\\gamma_{\\mu}\\bar\\eta_{1}=\n\\left(\\cos\\frac{\\varphi}{2}\\gamma_{1}-\\sin\\frac{\\varphi}{2}\\gamma_{2}\\right)\\bar\\eta_{1}=\\bar\\eta_{1}\\ \\ \\ \\ \\Rightarrow\\ \\ \\ \\ \n\\bar\\eta_{1\\alpha} =i\\begin{pmatrix} e^{i\\frac{\\varphi}{4}}\\\\ e^{-i\\frac{\\varphi}{4}}\\end{pmatrix}\n\\end{equation}\nas a result\\footnote{We have dropped a global phase in $\\bar\\eta_{1}$ since it can be eliminate through a global $U(1)$ redefinition of the matter fields.} of the  two constraints \\eqref{orto1}.\nSimilarly\nthe spinor $\\eta_{1}^{\\alpha}$ obeys the hermitian conjugate of the above equation and thus\n$\\eta_{1}^{\\alpha}\\propto (\\bar\\eta_{1\\alpha})^{\\dagger}$. The  condition \\eqref{norm}\nfixes the relative normalization and we find\n\\begin{equation}\n\\eta_{1}^{\\alpha}= ( e^{-i\\frac{\\varphi}{4}}\\ \\ \\  e^{i\\frac{\\varphi}{4}}).\n\\end{equation}\nOn the other hand\nthe $R-$symmetry part of the couplings   is arbitrary and in fact $n_{1M}$ and $\\bar n_{1}^{M}$ are totally unconstrained. For future convenience we  choose \n\\begin{equation}\nn_{1M}=\\mbox{\\small$ \\left(\\cos\\frac{\\theta}{4}\\ \\ \\sin\\frac{\\theta}{4}\\ \\ 0\\ \\ 0\\right)$}\n\\ \\  \\  \\ \\mathrm{and}\n\\ \\ \\ \\  \n\\bar n_{1}^{M}=\\mbox{\\footnotesize $\n\\begin{pmatrix}\\cos\\frac{\\theta}{4}\\\\ \\sin\\frac{\\theta}{4}\\\\ 0\\\\ 0\\end{pmatrix}$}.\n\\end{equation}\nOn the second edge, again as a result of  \\eqref{orto1},   $\\bar\\eta_{2}$ must be  the \neigenspinor of eigenvalue $1$ of $\\frac{\\dot{x}_{2}^{\\mu}}{|\\dot x_{2}|}\\gamma_{\\mu}$ and following the same route we get\n\\begin{equation}\n\\bar\\eta_{2\\alpha} =i e^{i\\delta}\\begin{pmatrix} e^{-i\\frac{\\varphi}{4}}\\\\ e^{i\\frac{\\varphi}{4}}\\end{pmatrix}\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\n\\eta_{2}^{\\alpha}=e^{-i\\delta} ( e^{i\\frac{\\varphi}{4}}\\ \\ \\  e^{-i\\frac{\\varphi}{4}}).\n\\end{equation}\nThe arbitrary phase $\\delta$ cannot be reabsorbed into a redefinition of the fields without altering the structure of the fermionic couplings on the first edge. For the $R-$symmetry sector in \\eqref{cuspcoupling}\nwe instead set \n\\begin{equation}\nn_{2M}=\\mbox{\\small$ \\left(\\cos\\frac{\\theta}{4}\\ \\ -\\sin\\frac{\\theta}{4}\\ \\ 0\\ \\ 0\\right)$}\n\\ \\  \\  \\ \\mathrm{and}\n\\ \\ \\ \\  \n\\bar n_{2}^{M}=\\mbox{\\footnotesize $\n\\begin{pmatrix}\\cos\\frac{\\theta}{4}\\\\ -\\sin\\frac{\\theta}{4}\\\\ 0\\\\ 0\\end{pmatrix}$}.\n\\end{equation}\nThe two matrices which couple the scalars are then determined through the relations \\eqref{cc}, which give\n$M$ and $\\widetilde{M}$ in terms of $n$ and $\\bar n$. On the two edges we have respectively\n\\begin{equation}\nM_{1J}^{\\ \\ I}=\n\\widehat M_{1J}^{\\ \\ I}=\\mbox{\\small $\\left(\n\\begin{array}{cccc}\n -\\cos \\frac{\\theta }{2}& -\\sin \\frac{\\theta }{2} & 0 & 0 \\\\\n -\\sin \\frac{\\theta }{2}& \\cos\\frac{\\theta }{2} & 0 & 0 \\\\\n 0 & 0 & 1 & 0 \\\\\n 0 & 0 & 0 & 1\n\\end{array}\n\\right)$}\\ \\ \\ \\ \\mathrm{and}\\ \\ \\ \\  M_{2J}^{\\ \\ I}=\n\\widehat M_{2J}^{\\ \\ I}=\\mbox{\\small $\\left(\n\\begin{array}{cccc}\n -\\cos \\frac{\\theta }{2} & \\sin \\frac{\\theta }{2} & 0 & 0 \\\\\n \\sin \\frac{\\theta }{2} & \\cos\\frac{\\theta }{2} & 0 & 0 \\\\\n 0 & 0 & 1 & 0 \\\\\n 0 & 0 & 0 & 1\n\\end{array}\n\\right)$}.\n\\end{equation}\n\\subsection{Intermediate BPS configurations}\n\\label{subsecBPS}\nIn the following we would like to explore if there is a choice of the  $(\\varphi,\\theta,\\delta)$ such \nthat the generalized cusp turns out to be  BPS\\footnote{Of course we have an obvious one: $\\varphi=\\theta=\\delta=0$.}.\nThese configurations may provide useful checks for the perturbative computations, but they can also provide\na tool for addressing the issue of nonperturbative computations \\cite{Correa:2012at}.\n\nLet us consider one of  the  Poincar\\`e  supersymmetries  preserved by the first edge of the cusp in fig. \\ref{cusp1}.\n As discussed in sec. \\ref{sec2},  it admits the following expansion\n \\begin{equation}\n \\label{charge1}\n \\bar\\theta_{1}^{IJ\\beta}= (\\bar\\eta_{1}^{I\\beta} \\bar v^{J}_{1}-\\bar\\eta_{1}^{J\\beta} \\bar v^{I}_{1})\n-i\\epsilon^{IJKL} \\eta_{1K}^{\\beta} u_{1L}=\n(\\bar n^{I}_{1} \\bar v^{J}_{1}-\\bar n^{J}_{1} \\bar v^{I}_{1})\\bar\\eta_{1}^{\\beta}\n-i\\epsilon^{IJKL} n_{1K} u_{1L}\\eta_{1}^{\\beta},\n\\end{equation}\n where $ \\eta_{1K}$   $\\bar\\eta_{1}^{I}$ are the spinor couplings on the first line. The choice of the two vectors $u_{1}$ and $\\bar v_{1}$ selects the  charge that  we are considering. We observe that if \\eqref{charge1} defines \na supercharge shared  with the second edge it must admit a similar  expansion in terms of the spinor couplings of  the second line. Expanding $\\theta^{IJ}_{1}$ in the basis provided by $\\eta_{2}$ and $\\bar\\eta_{2}$ , we obtain the following system of equation    \n\\begin{subequations}\n\\label{cond1}\n\\begin{align}\n-i\\epsilon^{IJKL} n_{2K} u_{2L}=&-i e^{i\\delta}\\epsilon^{IJKL} n_{1K} u_{1L}\\cos\\frac{\\varphi}{2}+(\\bar n^{I}_{1} \\bar v^{J}_{1}-\\bar n^{J}_{1} \\bar v^{I}_{1})e^{i\\delta} \\sin\\frac{\\varphi}{2}\\label{cond1a}\\\\\n(\\bar n^{I}_{2} \\bar v^{J}_{2}-\\bar n^{J}_{2} \\bar v^{I}_{2})=& ~i e^{-i\\delta}\\epsilon^{IJKL} n_{1K} u_{1L}\\sin\\frac{\\varphi}{2}+(\\bar n^{I}_{1} \\bar v^{J}_{1}-\\bar n^{J}_{1} \\bar v^{I}_{1})e^{-i\\delta} \\cos\\frac{\\varphi}{2}.\\label{cond1b}\n\\end{align}\n\\end{subequations}\nWhen  this set of equations can be consistently solved both for $u_{2}$ and $\\bar v_{2}$, we have found a \ncandidate BPS configuration.   To begin with, we shall multiply \\eqref{cond1a} by  $ n_{2J}$. The resulting\ncondition does not contain $u_{2}$ and $\\bar v_{2}$: it is actually a constraint on the super-charge $\\bar\\theta_{1}^{IJ}$\n\\begin{equation}\n\\label{consist1}\n\\epsilon^{IJKL}n_{2 J} n_{1K} u_{1L}\\cos\\frac{\\varphi}{2}+i \\left(\\bar n^{I}_{1} (n_{2J}\\bar v^{J}_{1})-\\cos\\frac{\\theta}{2} \\bar v^{I}_{1}\\right) \\sin\\frac{\\varphi}{2}=0.\n\\end{equation}\nIf we project  \\eqref{consist1} onto the direction $n_{1I}$, we have immediately\n\\begin{equation}\n(n_{2 J} \\bar v_{1}^{J}) \\sin\\frac{\\varphi}{2} e^{i\\delta}=0\\ \\ \\ \\ \\Rightarrow\\ \\ \\ \\  (n_{2 J} \\bar v_{1}^{J}) =0.\n\\end{equation}\nand consequently from \\eqref{consist1}\n\\begin{equation}\n\\label{v1}\n\\bar v^{I}_{1} =-i\\frac{\\cot\\frac{\\varphi}{2}}{\\cos\\frac{\\theta}{2}}\n\\epsilon^{IJKL}n_{2 J} n_{1K} u_{1L}.\n\\end{equation}\nNext we multiply  \\eqref{cond1b} by $\\epsilon_{IJMN} \\bar n_{2}^{M}$. Again the dependence on $u_{2}$ and $v_{2}$ drops out and we end up with the following constraint\n\\begin{equation}\n0= i\\left (\\cos\\frac{\\theta}{2} u_{1N}-n_{1N} (\\bar n_{1}^{M}u_{1M})\\right)\\sin\\frac{\\varphi}{2}+\\epsilon_{IJMN}\\bar n^{I}_{1} \\bar v^{J}_{1}\\bar n_{2}^{M}\\cos\\frac{\\varphi}{2},\n\\end{equation}\nwhich is equivalent to \n\\begin{equation}\n\\label{u1}\nu_{1 L}=i\\frac{\\cot\\frac{\\varphi}{2}}{\\cos\\frac{\\theta}{2}} \\epsilon_{RSML}\\bar n^{R}_{1} \\bar v^{S}_{1}\\bar n_{2}^{M}.\n\\end{equation}\nThe relations \\eqref{v1} and \\eqref{u1} are consistent  if and only if\n\\begin{equation}\n\\bar v^{I}_{1} =-i\\frac{\\cot\\frac{\\varphi}{2}}{\\cos\\frac{\\theta}{2}}\n\\epsilon^{IJKL}n_{2 J} n_{1K} u_{1L}=\\frac{\\cot^{2}\\frac{\\varphi}{2}}{\\cos^{2}\\frac{\\theta}{2}}\n\\epsilon^{IJKL}n_{2 J} n_{1K} \\epsilon_{RSML}\\bar n^{R}_{1} \\bar v^{S}_{1}\\bar n_{2}^{M}=\n\\frac{\\cot^{2}\\frac{\\varphi}{2}}{\\cot^{2}\\frac{\\theta}{2}} v_{1}^{I},\n\\end{equation}\nnamely $\\cot^{2}\\frac{\\varphi}{2}=\\cot^{2}\\frac{\\theta}{2}.$  \nTherefore for $\\theta=\\pm\\varphi$ we expect that  the loop operator defined in the previous section is  BPS.\nIn fact for this value of the parameters we can find an explicit and simple solution for $u_{2}$ and $\\bar v_{2}$\n\\begin{align}\nu_{2I}=&e^{i\\delta } u_{1I}\\ \\ \\mathrm{and} \\ \\  \\bar v^{I}_{2}= e^{-i\\delta} v^{I}_{1}.\n\\end{align}\nWe remark we still need another property to fully confirm the presence of a BPS configuration at $\\theta=\\pm\\varphi$: we should prove that the gauge functions $g_{1}$ and $\\bar g_{2}$ on the two edges define a globally well-defined gauge transformation, which is continuous when we cross the cusp.  The values of  $g_{1}$ \non the two edges are given by\n\\begin{equation}\n\\mbox{\\footnotesize \\sc [First Edge]}: 4i\\sqrt{\\frac{2\\pi}{k}}  \\bar v^{L}_{1}C_{L} \\ \\ \\  \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \n\\mbox{\\footnotesize  \\sc [Second Edge]}:4i\\sqrt{\\frac{2\\pi}{k}}\\bar v^{L}_{1}e^{-i\\delta}C_{L} ,\\\n\\end{equation}\nwhile for $\\bar g_{2}$ we find\n\\begin{equation}\n\\mbox{\\footnotesize \\sc [First Edge]}: 4\\sqrt{\\frac{2\\pi}{k}}  u_{1L}\\bar C^{L} \\ \\ \\  \\ \\ \\ \\ \\ \\ \\\n\\mbox{\\footnotesize  \\sc [Second Edge]}: 4\\sqrt{\\frac{2\\pi}{k}} e^{i\\delta} u_{1L}\\bar C^{L}.\n \\end{equation}\n Only  for $\\delta=0$ the two gauge function are continuous through the cusp.  Summarizing, \n for the $\\theta=\\pm \\varphi$ and $\\delta=0$ the generalized cusp of fig.\\ref{cusp1}\n is BPS. The preserved Poincar\\`e supercharges in terms of the quantity of the first line  can be then \n written in the following two equivalent ways\n  \\begin{align}\n  \\label{34}\n \\bar\\theta_{1}^{IJ\\beta}=&\n-i\\left(\\frac{\\bar n^{I}_{1} }{\\sin\\frac{\\varphi}{2}}\n\\epsilon^{JMKL}n_{2 M} n_{1K} u_{1L}-\\frac{\\bar n^{J}_{1} }{\\sin\\frac{\\varphi}{2}}\n\\epsilon^{IMKL}n_{2 M} n_{1K} u_{1L}\\right)\\bar\\eta_{1}^{\\beta}\n-i\\epsilon^{IJKL} n_{1K} u_{1L}\\eta_{1}^{\\beta}=\\nonumber\\\\\n =& \n(\\bar n^{I}_{1} \\bar v^{J}_{1}-\\bar n^{J}_{1} \\bar v^{I}_{1})\\bar\\eta_{1}^{\\beta}\n-\\frac{1}{\\sin\\frac{\\varphi}{2}}\\left(\\left(\\bar n^{I}_{2}- \\cos\\frac{\\varphi}{2} \\bar n^{I}_{1}\\right) \n\\bar v^{I}_{1}-\\left(\\bar n^{J}_{2}- \\cos\\frac{\\varphi}{2} \\bar n^{J}_{1}\\right) \n\\bar v^{I}_{1}\\right)\\eta_{1}^{\\beta}.\n\\end{align}\nThe vector  $ u^{I}_{1}$ in the first line  of \\eqref{34} and the vector $\\bar v^{I}_{1}$ in the second one\nmust obey  $ (n_{2 J} \\bar v_{1}^{J})=(n_{1 J} \\bar v_{1}^{J})=0$ and $ (\\bar n_{2 J} u_{1}^{J})=(\\bar n_{1 J} u_{1}^{J})=0$ respectively. Thus we have   two shared  Poincar\\`e supercharges. \n\nA remark on the conformal supercharges $\\bar\\vartheta^{IJ}$ is now in order: for each edge of the cusp they admit the same expansion \\eqref{susypar} which was obtained for the Poincar\\`e ones. The above analysis implies therefore that there are two shared superconformal charges as well.\n\n\n \\subsection{Mapping  the cusp to the spherical wedge}\n \\label{subsecwedge}\n \nRecently  \\cite{Correa:2012at}  it was noticed that the DGRT spherical wedge \\cite{Drukker:2007qr}, which is a BPS loop\n operator, can be used to extract nonperturbative information about the generalized cusp\n \\begin{wrapfigure}[14]{l}{68mm}\n\\vskip -.9cm\n    \\includegraphics[width=.40\\textwidth, height=.35 \\textwidth]{cusptowedge.pdf}\n\\vskip -.3cm\n\\caption{\\label{cusptowedge}The cusp in the plane $x^{0}=1\/2$ is mapped into the spherical wedge under the conformal transformation generated by the vector $(1,0,0)$.}\n\\end{wrapfigure}\n in $\\mathcal{N}=4$ SYM, since its value is known at all order in the coupling constant.  It was argued in \\cite{Drukker:2007qr} that the exact quantum result is obtained from the ordinary circular Wilson loop computation, with $\\lambda$  replaced by\n \\begin{equation}\n\\lambda \\mapsto \\tilde\\lambda=\\lambda\\frac{4 A_{1} A_{2}}{A^{2}},\n\\end{equation}\n where $A_{1}$ and $A_{2}$ are the areas of the two sides of the contour and $A = A_{1} + A_{2}$ is the total\narea of the two-sphere. Since the DGRT spherical wedge can be related to the BPS  configuration of the generalized cusp in $\\mathcal{N}=4$ through a special conformal transformation, it is tempting to \ninvestigate what is the image of the operator defined in subsect. \\ref{subseccoupl} when we map the plane\nsupporting it  onto a sphere $S^{2}$.\n \nOur starting point is not the  cusp in the plane $(1,2)$, as  in subsect. \\ref{subseccoupl}, but for simplicity one located   in the plane $x^{0}=1\/2$, {\\it i.e.}\n\\begin{equation}\n\\label{cusp3}\nx^{0}=1\/2 \n\\ \\ \\ \\ \\ \nx^{1}=s\\cos\\frac{\\varphi}{2} \\ \\ \\ \\ \\  x^{2}=|s|\\sin\\frac{\\varphi}{2} \\ \\ \\  \\  -\\infty\\le s,\\le \\infty.\n\\end{equation} \nThe  plane  $x^{0}=1\/2$ can be mapped into the usual unit sphere  centered in the origin through the special conformal transformation generated by the vector  $b^{\\mu}=(1,0,0)\n\\begin{equation}\n\\label{conf2}\n y^{0}=\\frac{x^{0}- x^{\\mu} x_{\\mu}}{1-2 x^{0}+ x^{\\mu} x_{\\mu}},\\ \\ \\ \\ \\ \n y^{1}=\\frac{x^{1}}{1-2x^{0}+ x^{\\mu} x_{\\mu}},\\ \\ \\ \\ \\ y^{2}=\\frac{x^{2}}{1-2x^{0}+ x^{\\mu} x_{\\mu}}.\n\\end{equation}\nThen  the image of the original contour is the wedge  illustrated in fig. \\ref{cusptowedge}.  The new path \nin terms of $s$ is given by \n\n\n\\begin{equation}\n\\label{wedges}\n\\!\\!\ny^{0}=\\frac{\\frac{1}{4}-s^{2}}{\\frac{1}{4}+s^{2}},\\ \\ y^{1}=\\frac{ s\\cos\\frac{\\varphi}{2}}{\\frac{1}{4}+s^{2}},\\ \\ \ny^{2}=\\frac{ |s|\\sin\\frac{\\varphi}{2}}{\\frac{1}{4}+s^{2}}.\n\\end{equation}\nWhen $s$ ranges from $-\\infty$ to $\\infty$ we move from  the south  of the sphere $(-1,0,0)$ to the north pole $(1,0,0)$ [$s=0$] and back to the south pole. The usual parametrization in polar coordinates is recovered by \nperforming the substitution  $s=\\frac{1}{2}\\tan\\frac{t}{2}$ in \\eqref{wedges}\n\\begin{equation}\n\\!\\!\ny^{0}\\!=\\!\\cos t,\\  y^{1}\\!=\\sin t\\cos\\frac{\\varphi}{2},\\ \ny^{2}=|\\sin t|\\sin\\frac{\\varphi}{2}\n\\end{equation}\nwhere $ t\\in[-\\pi,\\pi]$.\nThe effect of the change of coordinates \\eqref{conf2} on the fermionic couplings  $\\eta_{I}$ and $\\bar\\eta^{I}$ is more  interesting and straightforward  to  determine once we recall the result of a finite special conformal transformation on a spinor field\\footnote{In three dimensions, the finite form of a special conformal transformation on a spinor field is\n\\[\n\\psi^{\\prime }( y)=(1-2 (b\\cdot x) +b^{2} x^{2}) ^{1\/2}\n (\\mathds{1}-b^{\\mu}\\gamma_{\\mu} x^{\\alpha}\\gamma_{\\alpha})\\psi(x).\n \\] } (see {\\it e.g.} \\cite{Ho}). Comparing the anti-diagonal term of the matrix \\eqref{superconnection} in the two different coordinates, we find for instance\n \\begin{equation}\n |\\dot y| \\eta^{\\prime}_{I}\\bar\\psi^{\\prime I}( y)=|\\dot{x}|(1-2 (b\\cdot x) +b^{2} x^{2}) ^{-1\/2}\n \\eta^{\\prime }_{I}  (\\mathds{1}-b^{\\mu}\\gamma_{\\mu} x^{\\alpha}\\gamma_{\\alpha})\\bar\\psi^{I }(x)=\n |\\dot{x}|\\eta_{I}\\bar\\psi^{I}(x),\n \\end{equation}\n namely\n\\begin{equation}\n\\label{transfeta}\n  \\eta^{\\prime }_{I}  =\n \\eta_{I} \\frac{(\\mathds{1}- x^{\\alpha}\\gamma_{\\alpha}b^{\\mu}\\gamma_{\\mu})}{\\sqrt{1-2 (b\\cdot x) +b^{2} x^{2}}}.\n\\end{equation}\nIn other words, in the case of  the spinor couplings, the  effect of mapping the cusp into spherical  wedge  translates into a local rotation defined by the matrix appearing in \\eqref{transfeta}.  We have a different rotation on each edge\n\\begin{subequations}\n \\paragraph{\\sc Edge I:}  $(s<0)$\n \\begin{equation}\n  \\label{rotI}\nR_{I}=\\left(\n\\begin{array}{cc}\n \\frac{1}{\\sqrt{4 s^2+1}} & \\frac{2 e^{\\frac{i \\varphi }{2}} s}{\\sqrt{4 s^2+1}} \\\\\n -\\frac{2 e^{-\\frac{i \\varphi }{2}} s}{\\sqrt{4 s^2+1}} & \\frac{1}{\\sqrt{4 s^2+1}}\n\\end{array}\n\\right)=\\left(\n\\begin{array}{cc}\n \\cos \\frac{t}{2} & e^{\\frac{i \\varphi }{2}} \\sin\\frac{t}{2} \\\\\n -e^{-\\frac{i \\varphi }{2}} \\sin \\frac{t}{2} & \\cos \\frac{t}{2}\n\\end{array}\n\\right)\n\\end{equation}\n \\paragraph{\\sc Edge II:}  $(s>0)$\n \\begin{equation}\n \\label{rotII}\nR_{II}=\\left(\n\\begin{array}{cc}\n \\frac{1}{\\sqrt{4 s^2+1}} & \\frac{2 e^{-\\frac{i \\varphi }{2}} s}{\\sqrt{4 s^2+1}} \\\\\n -\\frac{2 e^{\\frac{i \\varphi }{2}} s}{\\sqrt{4 s^2+1}} & \\frac{1}{\\sqrt{4 s^2+1}}\n\\end{array}\n\\right)=\\left(\n\\begin{array}{cc}\n \\cos \\frac{t}{2} & e^{-\\frac{i \\varphi }{2}} \\sin \\frac{t}{2} \\\\\n -e^{\\frac{i \\varphi }{2}} \\sin\\frac{t}{2}& \\cos\\frac{t}{2}\n\\end{array}\n\\right).\n \\end{equation}\n \\end{subequations}\n \n We have expressed the rotation matrices   in terms of both the original parameter $s$ and  the parameter \n $t=2\\arctan(2 s)$.  Now  the fermionic couplings on the two sides of the wedge  are obtained by rotating the old ones  by means of the two matrices  $R_{I}$ and $R_{II}$\n \\begin{equation}\n \\upsilon_{1}^{\\alpha}=(\\eta_{1}R_{I})^{\\alpha}=\\mbox{\\small$\\left(e^{-i\\frac{\\varphi}{4}}\\left(\\cos \\frac{t}{2}-\\sin\\frac{t}{2}\\right)  \\  \\ e^{i\\frac{\\varphi}{4}}\\left(\\cos \\frac{t}{2}+\\sin\\frac{t}{2}\\right)\\right)$}\n \\end{equation}\n  \\begin{equation}\n \\upsilon_{2}^{\\alpha}=(\\eta_{2}R_{II})^{\\alpha}=\\mbox{\\small$\\left(e^{i\\frac{\\varphi}{4}}\\left(\\cos \\frac{t}{2}-\\sin\\frac{t}{2}\\right)  \\  \\ e^{-i\\frac{\\varphi}{4}}\\left(\\cos \\frac{t}{2}+\\sin\\frac{t}{2}\\right)\\right)$}\n \\end{equation}\n and obviously $\\bar\\upsilon_{1,2\\alpha}=i (\\upsilon_{1,2}^{\\alpha})^{\\dagger}$.  The  matrices $M$ and \n $\\widehat M$ which  couple the scalars to the loop are instead  unaffected by the  special conformal transformation. \n \n Next we consider the effect of the change of variables \\eqref{conf2} on  the  preserved  super-charges of subsect. \\ref{subsecBPS}.  The super-conformal Killing spinors  $(\\bar{\\theta}^{IJ},\\bar\\vartheta^{IJ})$   of the cusp    are  transformed into \n \\begin{equation}\n(\\bar{\\theta}^{IJ},\\bar\\vartheta^{IJ})\\mapsto \n(\\bar{\\theta}^{IJ},\\bar\\vartheta^{'IJ})=(\\bar{\\theta}^{IJ},\\bar\\vartheta^{IJ}+\\bar\\theta^{IJ}(b^{\\mu}\\gamma_{\\mu}))=\n(\\bar{\\theta}^{IJ},\\bar\\vartheta^{IJ}+\\bar\\theta^{IJ}\\gamma_{0}).\n\\end{equation}\nThe loop operator defined by this spherical wedge is preserved by the  conformal Killing spinors with a structure given by\n$\\bar\\Theta^{IJ}=\\bar\\theta^{IJ} (1+y^{\\mu}\\gamma_{0}\\gamma_{\\mu})+y^{\\mu}\\bar\\vartheta^{IJ}\\gamma_{\\mu}. $ \n\nIn doing the conformal transformation we have effectively compactified the contour and we have to understand what happens to the gauge functions at the south pole: continuity of the gauge transformations at north pole is instead inherited by the BPS properties of the open cusp. It  is a straightforward exercise to compute the spinor contractions which are relevant in determining the gauge function $g_{1}$ and $\\bar g_{2}$ at the points $t=-\\pi$ and  $t=\\pi$:\n\\begin{equation}\n\\label{ghfun}\n\\begin{array}{llll}\n\\left. g_{1}\\right|_{t=-\\pi}=   4 i\\sqrt{\\frac{2\\pi}{k}} (\\bar{\\tilde v}^{L}_{1}C_{L})\n& & &\n\\left. g_{1}\\right|_{t=\\pi}= \n  -4 i\\sqrt{\\frac{2\\pi}{k}} (\\bar{\\tilde v}^{L}_{2}C_{L})\\\\\n  & & &\\\\\n\\left.\\bar g_{2}\\right|_{t=-\\pi}=\n4 \\sqrt{\\frac{2\\pi}{k}}   (\\tilde u_{1L}\\bar C^{L})\n& & &\n\\left.\\bar g_{2}\\right|_{t=\\pi}=\n-4 \\sqrt{\\frac{2\\pi}{k}}   (\\tilde u_{2L}\\bar C^{L})\n\\end{array}\n\\end{equation} \nwhere we used that $\\vartheta^{IJ}$ admits  the same expansion  of the Poincar\\`e supercharges in terms of two vectors $\\tilde u_{i}$ and  $\\bar{\\tilde v}_{i}$. These last two vectors will obey the same constraint of $u_{i}$ and $\\bar v_{i}$ and in particular for the shared supercharges $\\bar{\\tilde v}^{I}_{1}=\\bar{\\tilde v}^{I}_{2}$ and \n$\\tilde u^{I}_{1}=\\tilde u^{I}_{2}$. We see the gauge functions \\eqref{ghfun} are anti-periodic and consequently, to have a BPS loop,  we have to take the trace to obtain a {\\it supersymmetric} wedge on $S^{2}$. This is consistent with the result of \\cite{Drukker:2009hy}, our wedge being a non-trivial BPS deformation of the BPS circle. It is interesting that within our construction the antiperiodicity of gauge functions appears as an effect of the conformal mapping, rather that being assumed from the beginning. The presence of such supersymmetric configurations suggests also that it should be possible to construct a general class of BPS loops on $S^2$, representing the ABJ analogue of the DGRT loops of ${\\cal N}=4$. The explicit construction and the quantum analysis of this new family, as well as of the analogue of Zarembo's loops in superconformal Chern-Simons theories, will be the subjects of a separate publication \\cite{CGMS}. \n\n\n\\section{Quantum results}\nWe shall compute the expectation value of the generalized {\\it cusp } operator up to the  second order in the coupling constant $\\left(\\frac{2\\pi}{\\kappa}\\right)$. So far there are very few results about the perturbative properties of supersymmetric Wilson loops in ABJ(M) theories and they are all strictly confined to the 1\/6 BPS {\\it bosonic} case \\cite{Drukker:2008zx,Chen:2008bp,Rey:2008bh}. We remark that even the matrix model \\cite{Drukker:2009hy} - believed  to capture the exact result for the 1\/2 BPS circle -  has not been verified by explicit Feynman diagrams computations.\n\nThe quantum holonomy of\nthe super-connection ${\\cal L}$\n in a representation ${\\cal R}$ of the supergroup $U(N|M)$ is by definition\n\n\\begin{equation}\n\\label{eq:loopexpectationvalue}\n\\left\\langle \\mathcal{W}_{\\cal R} \\right\\rangle= \\frac{1}{{\\rm dim}_{\\cal R}}\\int {\\cal D}[A,\\hat{A},C,\\bar{C},\\psi,\\bar{\\psi}]~\n{\\rm e}^{-S_{\\rm ABJ}}~{\\rm Tr}_{\\cal R} \\left[\n  {\\rm P} \\exp \\left(- i\\int_{\\Gamma} d\\tau\\, {\\cal L}(\\tau) \\right) \\right],\n\\end{equation}\nwhere $S_{ABJ}$ is the Euclidean action for $ABJ(M)$ theories (the part relevant for our computation is spelled out in app. \\ref{ABJ}).  In the following $\\mathcal{R}$ will be chosen to be the fundamental representation.\n\n In order to evaluate $\\left\\langle \\mathcal{W}_{\\cal R} \\right\\rangle$  we  shall first focus our attention  on the upper left $N\\times N$ block of the super-matrix appearing  in \\eqref{eq:loopexpectationvalue}.\n For this sub-sector the trace is obviously taken in the fundamental representation ${\\bf N}$ of the first gauge group. The result for the lower\n diagonal  block can be then recovered from this analysis by exchanging $N$ with $M$. Our perturbative computation  requires to expand the path-exponential   in \\eqref{eq:loopexpectationvalue} up to  the fourth order. The  terms in this expansion relevant for the upper block include both  {\\it bosonic} and {\\it fermionic} monomials:\n\\begin{align}\n\\label{expaloop}\n\\mathbb{W}_{\\mathbf{N}}&={\\rm Tr}_{\\mathbf{N}}\\left[1+i\\int_\\Gamma d\\tau_1{\\cal A}_1-\\int_{\\Gamma}d\\tau_{\\mbox{\\tiny $\\displaystyle1\\!\\!>\\!\\! 2$}}\\Biggl({\\cal A}_1{\\cal A}_2-(\\eta\\bar{\\psi})_1(\\psi\\bar{\\eta})_2 \\Biggr) \\right.\\nonumber\\\\\n&-i\\int_{\\Gamma}d\\tau_{\\mbox{\\tiny $\\displaystyle1\\!\\!>\\!\\! 2\\!\\!>\\!\\!3$}}\\Biggl( {\\cal A}_1{\\cal A}_2{\\cal A}_3+\\frac{2\\pi}{k}[(\\eta\\bar{\\psi})_1(\\psi\\bar{\\eta})_2{\\cal A}_3 \n+(\\eta\\bar{\\psi})_1\\hat{\\cal A}_2 (\\psi\\bar{\\eta})_3+{\\cal A}_1(\\eta\\bar{\\psi})_2(\\psi\\bar{\\eta})_3]\\Biggr)\\nonumber\\\\\n&\\left.+\\int_{\\Gamma}d\\tau_{\\mbox{\\tiny $\\displaystyle1\\!\\!>\\!\\! 2\\!\\!>3\\!\\!>4$}}\\left(\\left(\\frac{2\\pi}{\\kappa}\\right)^{2}(\\eta\\bar{\\psi})_1(\\psi\\bar{\\eta})_2(\\eta\\bar{\\psi})_3(\\psi\\bar{\\eta})_4+\\mathcal{A}_{1}\\mathcal{A}_{2}\\mathcal{A}_{3}\\mathcal{A}_{4}-\\right.\\right.\\\\ \n&-\\left(\\frac{2\\pi}{\\kappa}\\right)\\mathcal{A}_{1}\\mathcal{A}_{2}(\\eta\\bar{\\psi})_3(\\psi\\bar{\\eta})_4-\\left(\\frac{2\\pi}{\\kappa}\\right)\\mathcal{A}_{1}(\\eta\\bar{\\psi})_2\\hat{\\mathcal{A}}_{3}(\\psi\\bar{\\eta})_4-\\left(\\frac{2\\pi}{\\kappa}\\right)(\\eta\\bar{\\psi})_1\\hat {\\mathcal{A}_{2}}\\hat{\\mathcal{A}_{3}}(\\psi\\bar{\\eta})_4-\\nonumber\\\\\n&\\left.\\left.\n-\\left(\\frac{2\\pi}{\\kappa}\\right)\\mathcal{A}_{1}(\\eta\\bar{\\psi})_2(\\psi\\bar{\\eta})_3\\mathcal{A}_{4}-\\left(\\frac{2\\pi}{\\kappa}\\right)(\\eta\\bar{\\psi})_1\\hat{\\mathcal{A}}_{2}(\\psi\\bar{\\eta})_3\\mathcal{A}_{4}-\\left(\\frac{2\\pi}{\\kappa}\\right)(\\eta\\bar{\\psi})_1(\\psi\\bar{\\eta})_2\\mathcal{A}_{3}\\mathcal{A}_{4}\n\\right)\\right].\\nonumber\n\\end{align}\nIn \\eqref{expaloop} we have introduced a shorthand notation for the circuit parameter dependence of the fields, namely ${\\cal A}_i = {\\cal A}(x_i)$ with $x_i = x(\\tau_i)$. We have also suppressed the spinor and $SU(4)_R$ indices and choosen the parametrization with $|\\dot{x}|=1$. The expression above is not symmetric in the exchange of the two gauge groups: the\n symmetry between them will be recovered when considering also the contribution coming from the lower right $M\\times M$ block.\n\n\\subsection{One-loop analysis}\n\\label{Leadord}\nThe first non-trivial contributions are proportional to $\\left(\\frac{2\\pi}{\\kappa}\\right)$ and involve both bosonic and fermionic diagrams. They are listed in fig. \\ref{fig:oneloopgraphs}.  Differently from what occurs for  the  ${\\cal N}=4$ generalized cusp, the  diagrams  which involve only one edge do not vanish when we add them up. The situation is a little more intricate and it is actually convenient to deal with them separately, also in view of the two-loop computation.\n\nThe evaluation of the  diagrams in fig. \\ref{fig:oneloopgraphs} obviously encounter UV divergences which originate  from the part of the integration region where the propagator endpoints  coincide. To tame these \ndivergences  we will extensively  use dimensional regularization. However regularizing Chern-Simons-matter theories going\noff-dimensions raises some concerns because of the presence of the anti-symmetric $\\epsilon^{\\mu\\nu\\rho}$ tensor.\n We will follow the DRED scheme, shifting the dimension to $d=3-2\\epsilon$ while keeping the Dirac algebra and $\\epsilon^{\\mu\\nu\\rho}$\ntensor strictly  in 3 dimensions. Note that this breaks the conformal invariance introducing a mass scale $\\mu^{2\\epsilon}$ that keeps the action dimensionless. We will also need an explicit IR regulator $L$, representing the finite length of the two rays forming the cusp: because of the underlying conformal invariance we expect that it could be always scaled away, combining into the $(\\mu L)^{2\\epsilon}$ to some powers that weights the relevant Feynman integrals.\n \\begin{figure}[ht]\n\\centering{\n    \\includegraphics[width=.97\\textwidth, height=.15\\textwidth]{1loopBosonicDiagrams.pdf}}\n\\vskip -.3cm\n\\caption{\\label{fig:oneloopgraphs}At one-loop order there are only two classes of diagrams: single-edge diagrams [(a), (b) and (c)] and  exchange diagrams [(d) and (e)]. The scalar field enters the loop operator only through the composite bilinear $M^I_J C_I \\bar{C}^J$, and its conjugate $\\widehat{M}^I_J \\bar{C}^J C_I$, hence the exchange of a single scalar is not permitted\n}\n\\end{figure}\n\nWe start by considering the bosonic diagrams:  the scalar tadpole of fig. \\ref{fig:oneloopgraphs}.(c), that \noriginates from the first term in the expansion  \\eqref{expaloop},  vanishes in our regularization procedure and we can safely forget its existence in the following. The  bosonic contributions in  fig. \\ref{fig:oneloopgraphs}.(a) and \\ref{fig:oneloopgraphs}.(d)  stems from the term with two gauge fields in \\eqref{expaloop} and they can be cast  into a single path-ordered integral,\n\\begin{equation}\n\\label{eq:gluon1loop}\n \\mathfrak{B}^{(1)}=\n {\\rm i} N^2\\left( \\frac{\\mu^{2\\epsilon}}{\\kappa}\\right)\n\\frac{\\Gamma(\\frac{3}{2}-\\epsilon)}{\\pi^{\\frac{1}{2}-\\epsilon}}\\,\\int_{\\Gamma}d\\tau_{\\mbox{\\tiny $\\displaystyle1\\!\\!>\\!\\! 2$}}\\, \n\\frac{\\epsilon_{\\mu\\nu\\rho} \\dot{x}_1^\\mu \\dot{x}_2^\\nu\n(\\dot{x}_1 -\\dot{x}_2 )^\\rho}{|x_1-x_2|^{3-2\\epsilon}}=0,\n\\end{equation}\nwhose integrand vanishes for any planar loop due to the antisymmetry of the $\\epsilon-$tensor\\footnote{In general one should also take into account the possibility to have framing contribution \\cite{Witten:1988hf}. We assume here that our computation can be consistently done at zero framing.}. We have used here the explicit form of the Chern-Simons propagator in position space, presented in App. \\ref{FRSS}. We remark that the same result would have been obtained if we have used 1\/6 BPS lines, in spite of the different structure of the mass-matrix $M^I_J$.\n\n\\noindent\nNext we discuss the fermionic diagrams  in fig. \\ref{fig:oneloopgraphs}.  They represent the true  novelty of the present  calculation and originate from taking the vacuum expectation value of the fermionic bilinear in the first line of (\\ref{expaloop})\n\\begin{equation}\n\\label{K1}\n \\mathfrak{F}^{(1)}=\\left(\\frac{2\\pi}{\\kappa} \\right)\\mu^{2\\epsilon}\\int_{\\tau_{2}<\\tau_{1}}\\!\\!\\!\\!\\! d\\tau_{1}d\\tau_{2}~\\langle \\mathrm{Tr}_{\\bf N}\\left[(\\eta\\bar\\psi)_{1} (\\psi\\bar\\eta)_{2} \\right]\\rangle=\\left(\\frac{2\\pi}{\\kappa} \\right)\\mu^{2\\epsilon}({\\cal F}_{b}+{\\cal F}_{e}),\n\\end{equation}\nwhere we have denoted with ${\\cal F}_{b}$ and ${\\cal F}_{e}$ the contributions corresponding to  the graphs  \\ref{fig:oneloopgraphs}.(b) and  \\ref{fig:oneloopgraphs}.(e) respectively. At the lowest order, the vacuum expectation value in \\eqref{K1} is simply obtained by contracting the fermion propagator \\eqref{fermionpropagator} with the\nspinors $\\eta$ and $\\bar\\eta$. We find \n\\begin{equation}\n\\label{contra}\n\\langle \\mathrm{Tr}_{\\bf N}\\left[(\\eta\\bar\\psi)_{1} (\\psi\\bar\\eta)_{2} \\right]\\rangle=i M N \\frac{\\Gamma(1\/2-\\epsilon)}{4\\pi^{3\/2-\\epsilon}}\n\\eta_{L1}\\gamma^{\\mu}\\bar\\eta_{2}^{L}\\partial_{x^\\mu_{1}}\\left(\\frac{1}{(x_{12}^{2})^{1\/2-\\epsilon}}\\right),\\end{equation}\nwhere  $x_{ij}=x_i-x_j$. The fermion bilinear $\\eta_{L1}\\gamma_{\\mu}\\bar\\eta_{2}^{L}$ can be readily evaluated for a general contour (and for general parametrization), thanks to the factorized form \\eqref{cc} of the spinor couplings and to the identity (\\ref{rg6}). We have\n\\begin{equation}\n\\label{rg6a}\n(\\eta_{L1}\\gamma^{\\mu}\\bar\\eta_{2}^L)=-2\\frac{(n_{1}\\cdot\\bar n_{2})}{(\\eta_{2}\\bar\\eta_{1})}\\left[\\frac{\\dot{x_{1}}^{\\mu}}{|\\dot x_{1}|}+\\frac{\\dot{x_{2}}^{\\mu}}{|\\dot{x}_{2}|}+\\frac{1}{2}\\frac{\\dot{x_{1}}^{\\lambda}}{|\\dot{x}_{1}|}\n\\frac{\\dot{x_{2}}^{\\nu}}{|\\dot x_{2}|}{\\rm i}\\epsilon^{\\lambda\\mu\\nu} \\right].\n\\end{equation}\nSince $\\dot{x}_{1}$, $\\dot{x}_{2}$ and $x_{12}=-x_{21}$ lay on the same plane, we can drop the wedge product in \\eqref{rg6a} and we obtain the  following  result\n\\begin{equation}\n\\label{propla}\n\\langle \\mathrm{Tr}_{\\bf N}\\left[(\\eta\\bar\\psi)_{1} (\\psi\\bar\\eta)_{2} \\right]\\rangle=-\\frac{i M N \\Gamma(1\/2-\\epsilon)}{2\\pi^{3\/2-\\epsilon} (\\eta_{2}\\bar\\eta_{1})}(n_{L1}\\bar n^{L}_{2})\n\\left[\\frac{\\dot x_{1}^\\mu}{|\\dot{x_{1}} | }+\\frac{\\dot{x_{2}}^{\\mu}}{|\\dot x_{2}|}\\right]\\partial_{x_{1}^{\\mu}}\\left(\\frac{1}{(x_{12}^{2})^{1\/2-\\epsilon}}\\right),\n\\end{equation}\nwhich holds  for any planar circuit.\nLet us specialize (\\ref{K1}) to the diagram of \\ref{fig:oneloopgraphs}(e): within our  parametrization of the circuit,  we can rearrange the effective propagator exchanged through the rays as the difference of two total derivatives\n\\begin{equation}\n\\left[\\frac{\\dot x_{1}^\\mu}{|\\dot{x_{1}}|}+\\frac{\\dot{x_{2}}^{\\mu}}{|\\dot x_{2}|}\\right]\\partial_{x_{1}^{\\mu}}\\left(\\frac{1}{(x_{12}^{2})^{1\/2-\\epsilon}}\\right)=\\left(\\frac{d}{d \\tau_{1}}-\\frac{d}{d \\tau_{2}}\\right)\\frac{1}{(\\tau_{1}^{2}+\\tau_{2}^{2}-2 \\tau_{1} \\tau_{2}\\cos\\varphi)^{1\/2-\\epsilon}}.\n\\end{equation}\nThe integration over the two edges can be done in a rather trivial way\n\\begin{equation}\n\\label{CD0}\n\\begin{split}\n&\\int_{0}^{L} d\\tau_{1}\\int_{-L}^{0} d\\tau_{2}\\left(\\frac{d}{d \\tau_{1}}-\\frac{d}{d \\tau_{2}}\\right)\\frac{1}{(\\tau_{1}^{2}+\\tau_{2}^{2}-2 \\tau_{1} \\tau_{2}\\cos\\varphi)^{1\/2-\\epsilon}}=\\\\\n&=-\\frac{L^{2\\epsilon}}{\\epsilon}+2 L^{2\\epsilon}\\int_{0}^{1} d \\tau \\frac{1}{(\\tau^{2}+2  \\tau  \\cos\\varphi+1)^{1\/2-\\epsilon}}.\n\\end{split}\n\\end{equation}\nThe remaining integral in \\eqref{CD0} is finite as $\\epsilon\\to0$ and it can be evaluated in terms of hypergeometric functions. However its  exact value for arbitrary $\\epsilon$ will not  be relevant for us and we shall only give its expansion around $\\epsilon=0$ at the lowest order\n\\begin{equation}\n\\int_{0}^{1} d \\tau \\frac{1}{(\\tau^{2}+2  \\tau  \\cos\\varphi+1)^{1\/2-\\epsilon}}=\\log \\left(\\sec \\left(\\frac{\\varphi }{2}\\right)+1\\right)+O(\\epsilon).\n\\end{equation}\nWe end up with\n\\begin{equation}\n{\\cal F}_{e}=\\frac{i M N \\Gamma(1\/2-\\epsilon)}{2\\pi^{3\/2-\\epsilon} (\\nu_{1}\\bar\\nu_{2})}(n_{L2}\\bar n^{L}_{1})L^{2\\epsilon}\\left[\\frac{1}{\\epsilon}-2\\log \\left(\\sec \\left(\\frac{\\varphi }{2}\\right)+1\\right)\\right]\n\\end{equation}\nand since $(n_{L2}\\bar n^{L}_{1})=\\cos\\frac{\\theta}{2}$ and $(\\nu_{1}\\bar\\nu_{2})=2i \\cos\\frac{\\varphi}{2}$ we get\n\\begin{equation}\n{\\cal F}_{e}=M N\\left(\\frac{ \\Gamma(1\/2-\\epsilon)}{4\\pi^{3\/2-\\epsilon}}\\right)\n\\frac{\\cos\\frac{\\theta}{2}}{\\cos\\frac{\\varphi}{2}}L^{2\\epsilon}\\left[\\frac{1}{\\epsilon}-2\\log \\left(\\sec \\left(\\frac{\\varphi }{2}\\right)+1\\right)\\right].\n\\end{equation}\nNext we must consider the case where the fermionic propagator connects two points on the same edge of the cusp, {\\it i.e.} the diagrams (b) in fig. \\ref{fig:oneloopgraphs}.  We have two  mirror graphs: one for each edge. The result of the first one  is provided by \n\\begin{equation}\n\\label{sed1l}\n-\\frac{ M N \\Gamma(1\/2-\\epsilon)}{4\\pi^{3\/2-\\epsilon}}\\int_{-L}^{0}\\!\\!\\! d\\tau_{1} \\int_{-L}^{\\tau_{1}} \\!\\! d\\tau_{2}\\left(\\frac{d}{d \\tau_{1}}-\\frac{d}{d \\tau_{2}}\\right)\\frac{1}{(\\tau_{1}-\\tau_{2})^{1-2\\epsilon}}=-\\frac{ M N \\Gamma(1\/2-\\epsilon)}{4\\pi^{3\/2-\\epsilon}} \\frac{L^{2\\epsilon}}{\\epsilon},\n\\end{equation}\nwhile the contribution of the second one simply doubles  \\eqref{sed1l}  and it yield\n\\begin{equation}\n{\\cal F}_b=-2\\frac{ M N \\Gamma(1\/2-\\epsilon)}{4\\pi^{3\/2-\\epsilon}} \\frac{L^{2\\epsilon}}{\\epsilon}.\n\\end{equation}\nTherefore the   complete  one loop  result for the upper left block  can be written as\n\\begin{equation}\\label{unsub1loop}\n\\mathfrak{F}^{(1)}=\\left(\\frac{2\\pi}{\\kappa}\\right)M N\\left(\\frac{ \\Gamma(1\/2-\\epsilon)}{4\\pi^{3\/2-\\epsilon}}\\right)\n(\\mu L)^{2\\epsilon}\\left[\\frac{1}{\\epsilon}\\left(\\frac{\\cos\\frac{\\theta}{2}}{\\cos\\frac{\\varphi}{2}}-2\\right)-2\\frac{\\cos\\frac{\\theta}{2}}{\\cos\\frac{\\varphi}{2}}\\log \\left(\\sec \\left(\\frac{\\varphi }{2}\\right)+1\\right)\\right].\n\\end{equation}\nThis result may appear  surprising at a first sight: while we could have expected the divergence from the cusp diagram (e), we have also a non-trivial contribution from the propagators living on a single edge (b). In ${\\cal N}=4$ SYM theory the analogous contributions, coming from the combined gauge-scalar propagator, are identically zero in Feynman gauge, and their potential divergence never enters into the game. Moreover in the limit $\\varphi=\\theta=0$ a non-vanishing and divergent result persists, contradicting the naive expectation that the BPS infinite line is trivial. \nTo understand the  result \\eqref{unsub1loop} and to extract from it the truly gauge-invariant cusp divergence, we have to recall some basics about the renormalization of (cusped) Wilson loops in gauge theories and to adapt the general procedure to our somehow exotic operators: this will be done in the next section, after having completed the two-loop computation.\n\nThe full one-loop expression is recovered by considering also the part coming from the lower $M\\times M$ block of the super-holonomy: it turns out to be the same, because of the symmetry between $N$ and $M$ at this order. The trace is simply obtained by adding this second contribution.\n\n\\section{Two-loop analysis}\nWe shall compute here the second order contribution to the expectation value of the cusped Wilson loop: we separate the computations of purely bosonic diagrams from fermionic ones, to appreciate technical and conceptual differences.\n\\subsection{Bosonic diagrams}\nWhen expanding the Wilson loop operator  at the second order  in the coupling constant, we encounter  the four families of merely bosonic contributions depicted in fig. \\ref{BosonicDiagrams}. We consider first the diagrams containing  the  one-loop corrected   gluon propagators (fig.~\\ref{BosonicDiagrams}.(a)). As we did in the one-loop analysis, \n we shall focus our  attention  on the upper diagonal block of the super-matrix, {\\it i.e.} on the $U(N)$ sector. With the help of \\eqref{oneloopgauge}, where the one-loop propagator is presented, we can immediately write\n\\begin{equation}\n\\label{gaugetwo-loop}\n\\begin{split}\n\\left[\\ref{BosonicDiagrams}.(a)\\right]_{\\rm up}\\!=\\!-M N^{2}\\left(\\frac{2\\pi}{\\kappa}\\right)^{2}\\frac{\\Gamma^{2}\\left(\\frac{1}{2}-\\epsilon\\right)}{4 \\pi^{3-2\\epsilon}}\\int_{\\Gamma} d\\tau_{\\mbox{\\tiny $\\displaystyle1\\!\\!>\\!\\! 2$}} \\! \\left[\\frac{(\\dot{x}_{1}\\cdot \\dot{x}_{2})}{((x-y)^{2})^{{1}-2\\epsilon}} -\\partial_{\\tau_{1}}\\partial_{\\tau_{2}}\\!\\frac{((x_{1}-x_{2})^{2})^{\\epsilon}}{4\\epsilon(1+2 \\epsilon)}\\right]\\!.\n\\end{split}\n\\end{equation}\nA similar structure is obtained when considering the correlator of  two scalar composite operators $M_{I}^{\\ J} C_{J}\\bar C^{I}$ in the diagram \\ref{BosonicDiagrams}.(b):\n\\begin{equation}\n\\label{scalartwo-loop}\n\\begin{split}\n\\left[\\ref{BosonicDiagrams}.(b)\\right]_{\\rm up}=M N^{2}\\left(\\frac{2\\pi}{\\kappa}\\right)^{2}\\frac{\\Gamma^{2}\\left(\\frac{1}{2}-\\epsilon\\right)}{16 \\pi^{3-2\\epsilon}}\\int_{\\Gamma} d\\tau_{\\mbox{\\tiny $\\displaystyle1\\!\\!>\\!\\! 2$}} \\  \\frac{|\\dot x_{1}||\\dot x_{2}|{\\rm Tr}(M_{1} M_{2})}{((x-y)^{2})^{{1}-2\\epsilon}} .\n\\end{split}\n\\end{equation}\nThe integrals  \\eqref{gaugetwo-loop} and  \\eqref{scalartwo-loop}   can be naturally combined together  to give\n\\begin{equation}\n\\label{totBos}\n\\!\n-M N^{2}\\left(\\frac{2\\pi}{\\kappa}\\right)^{\\!\\!2}\\!\\frac{\\Gamma^{2}\\!\\left(\\frac{1}{2}-\\epsilon\\right)}{4 \\pi^{3-2\\epsilon}}\\!\\int_{\\Gamma} \\!d\\tau_{\\mbox{\\tiny $\\displaystyle1\\!\\!>\\!\\! 2$}} \\! \\left[\\frac{(\\dot{x}_{1}\\!\\cdot \\!\\dot{x}_{2})-\\frac{1}{4}|\\dot x_{1}||\\dot x_{2}|{\\rm Tr}(M_{1} M_{2}) }{((x-y)^{2})^{{1}-2\\epsilon}} -\n\\partial_{\\tau_{1}}\\partial_{\\tau_{2}}\\frac{((x_{1}-x_{2})^{2})^{\\epsilon}}{4\\epsilon(1+2 \\epsilon)}\\right]\\!\\!.\n\\end{equation}\n \\begin{figure}[ht]\n\\centering{\n    \\includegraphics[width=.97\\textwidth, height=.14\\textwidth]{BosonicDiagrams.pdf}}\n\\vskip -.3cm\n\\caption{\\footnotesize\\label{BosonicDiagrams}\\hskip -0.1cm  Two loops bosonic diagrams:  (a) One-loop corrected  gauge propagators; (b)  Correlators of two composite scalar operators;  (c)  Correlators gauge field composite scalar operator; \n(d) Chern-Simons vertex diagrams; (e) Gluon double exchange diagrams.}\n\\end{figure}\nThe result \\eqref{totBos} deserves some comments. The last term in \\eqref{totBos} is a total derivative  and it\nwould correspond to a gauge transformation -albeit a singular one. In dimensional regularization it yields a \n$(\\theta,\\varphi)$ independent pole in $\\epsilon$ plus finite terms, thus its contribution to the  divergent  part of the cusp  becomes ineffective when we impose the renormalization condition discussed in subsec. \\ref{Leadord}.  The other contribution in \\eqref{totBos}, as firstly noted in \\cite{Drukker:2008zx},  possesses an unforeseen four-dimensional structure. When the two endpoints  lie on the same edge  it is proportional to  the  the tree-level effective propagator in $\\mathcal{N}=4$ since $\\mathrm{Tr}(M_{1} M_{2})=4$ and thus it vanishes. If they lie  instead  on opposite edges we get the following result\n\\begin{equation}\n\\label{Bos2loop}\n\\mathfrak{B}^{(2)}=\n-M N^{2}\\left(\\frac{2\\pi}{\\kappa}\\right)^{\\!\\!2}\\!\\frac{\\Gamma^{2}\\!\\left(\\frac{1}{2}-\\epsilon\\right)}{4 \\pi^{3-2\\epsilon}}\\left(\\cos\\varphi-\\cos^{2}\\frac{\\theta}{2}\\right )\\!\\int_{0}^{L} \\!\\!\\!d\\tau_{1}  \\int_{-L}^{0} \\!\\!\\!d\\tau_{2}  \\frac{1}{((x-y)^{2})^{{1}-2\\epsilon}} ,\n\\end{equation}\nwhere the integral governing the divergence is the same of the four dimensional case when we replace $2\\epsilon$ with $\\epsilon$.\n\nNext we examine the graphs \\ref{BosonicDiagrams}.(c), \\ref{BosonicDiagrams}.(d) and \\ref{BosonicDiagrams}.(e). The last one is identically zero for the same reasons of the one-loop\nsingle exchange \\ref{fig:oneloopgraphs}.(a). The diagram \\ref{BosonicDiagrams}.(c) for the case of \nplanar loop was discussed in \\cite{Drukker:2008zx} where it was found to vanish.  The same fate \nis shared by \\ref{BosonicDiagrams}.(d) as pointed out in \\cite{Henn:2010ps}. The only \ncontribution originating from the bosonic diagrams is therefore provided by \\eqref{Bos2loop}.\n\n\n\n \\subsection{Fermionic diagrams}\n\n\\begin{wrapfigure}[7]{l}{65mm}\n\\centering{\n    \\includegraphics[width=.23\\textwidth, height=.13\\textwidth]{FermionicBubble.pdf}}\n\\vskip -.3cm\n\\caption{\\label{2loopferm1} One-loop corrected fermions propagators}\n\\end{wrapfigure}\nThe simplest fermionic diagram appearing at the second order in perturbation theory\n consists of the \nexchange of the one-loop corrected fermion propagator depicted in fig \\ref{2loopferm1}.\n\nThe one-loop two-point  function for the spinor fields is  briefly discussed in app. \\ref{FRSS}. \nRemarkably  it again displays the  four dimensional behaviour already encountered in the bosonic\ncase.  Its form, in the DRED scheme, is \n\\begin{equation}\n\\left\\langle\n(\\psi_{I})_{\\hat i}^{\\  j}(x)(\\bar\\psi^{J})_{ k}^{\\ \\hat l}(y)\\right\\rangle^{1~\\rm \\ell oop}_{0}=\n-i\\left(\\frac{2\\pi}{\\kappa}\\right)\\delta_{\\hat i }^{\\\\\\hat l}\\delta^{j}_{k}(N-M)\\frac{\\Gamma ^{2}\\left(\\frac{1}{2}-\\epsilon \\right)}{16 \\pi ^{3-2 \\epsilon} }\\frac{1}{((x-y)^{2})^{1-2\\epsilon}}.\n\\end{equation}\nThe contribution to the upper block of the Wilson loop takes the following form\n\\begin{equation}\n\\begin{split}\n\\left(\\frac{2\\pi\\mu^{2\\epsilon}}{\\kappa}\\right)^{2} i MN(N-M) \\frac{\\Gamma ^{2}\\left(\\frac{1}{2}-\\epsilon \\right)}{16 \\pi ^{3-2 \\epsilon} }\\int_{\\Gamma}d\\tau_{\\mbox{\\tiny $\\displaystyle1\\!\\!>\\!\\! 2$}}  \\frac{(\\eta_{I1}\\bar\\eta^{I}_{2})}{((x_{1} -x_{2})^{2})^{1-2\\epsilon}}\n\\end{split}\n\\end{equation}\n\\subsubsection{Double Exchanges}\n\\label{DED}\nWe come now to discuss a more subtle  group of  diagrams, namely those involving two $\\langle\\psi\\bar\\psi\\rangle$ propagators. They arise when we evaluate  the contribution of the fermionic quadrilinear in \\eqref{expaloop}. At this order  its expansion yields  only two sets of  non-vanishing  Wick-contractions, weighted  by different group factor, and thus we arrive at the following integral\n\\begin{equation}\n\\label{DoubleExchange}\n\\begin{split}\n&-4\\left(\\frac{2\\pi\\mu^{2\\epsilon}}{\\kappa}\\right)^{\\!\\!2}\\!\\int_{\\Gamma}\\!\\!d\\tau_{\\mbox{\\tiny $\\displaystyle1\\!\\!>\\!\\! 2\\!\\!>\\!\\!\n3 \\!\\!>\\!\\! 4$}}[M^{2} N \\underset{(A1)}{S(x_{2}-x_{1}) S(x_{4}-x_{3})}-N^{2} M \\underset{(B1)}{S(x_{2}-x_{3}) S(x_{4}-x_{1})}].\n\n\\end{split}\n\\end{equation}\nHere the function $S(x_{i}-x_{j})$ is proportional to the two-point fermion correlator already encountered in \\eqref{contra} and it can be conveniently written as\n\\begin{equation}\nS(x_{i}-x_{j})=\\frac{(n_{i}\\cdot n_{j})}{(\\eta_{j}\\bar\\eta_{i})}(\\partial_{\\tau_{i}}-\\partial_{\\tau_{j}})D(x_{i}-x_{j}),\n\\end{equation}\n \\begin{wrapfigure}[10]{l}{92mm}\n\\centering{\n    \\includegraphics[width=.54\\textwidth, height=.24\\textwidth]{DoubleExchange.pdf}}\n\\vskip -.3cm\n\\caption{\\label{DoubleExc1class} First group of double-exchange diagrams}\n\\end{wrapfigure}\nwhere $D(x_{i}-x_{j})$ is the free scalar propagator  defined in \\eqref{scal1} while the couple of vec\\-tors $(n_{iI}, \\bar n_{i}^{I})$ and of  spinors $(\\eta_{Ii},\\bar\\eta_{ i}^{I})$ are defined in sec. \\ref{subseccoupl}. We shall consider  the two\ncontributions in  \\eqref{DoubleExchange} separately. In order to evaluate the term (A1) we have to \nsplit the region of integration in five  sectors that correspond to the five different  Feynman diagrams\ndepicted in fig. \\ref{DoubleExc1class}. Luckily we do not have to compute all of them. In fact graphs,\nwhich are related by a reflection with respect to the axis bisecting the cusp, yield the same result\\footnote{This equality can be shown by performing the change of variable  $s_{i}\\mapsto -s_{5-i}$ $(i=1,\\dots 4)$ and subsequently by restoring the integration in the canonical order.}. In other\nwords, the following equalities hold among the diagrams of fig.\\ref{DoubleExc1class}:  \\ref{DoubleExc1class}.(a)=\\ref{DoubleExc1class}.(e) and  \\ref{DoubleExc1class}.(b)=\\ref{DoubleExc1class}.(d). Moreover the graph\n\\ref{DoubleExc1class}.(c) is simply the square of  \\ref{fig:oneloopgraphs}.(b).\n\nTo begin with, let us evaluate  the contribution \\ref{DoubleExc1class}.(a).  It is given by the following integral\n\\begin{align}\n[\\ref{DoubleExc1class}.(a)]_{\\rm up\n\\!=\\!&\\left(\\frac{2\\pi}{\\kappa}\\right)^{\\!\\!2}\\!\\!M^{2}\\!N\\frac{\\Gamma^{2}\\left(\\frac{3}{2}-\\epsilon\\right)}{ \\pi^{{3}-2\\epsilon}}\\mu^{4\\epsilon}\\!\\!\\int_{-L}^{0}\\!\\!\\!\\!\\!d\\tau_{1}\\int_{-L}^{\\tau_{1}}\\!\\!\\!\\!\\!d\\tau_{2}\\int_{-L}^{\\tau_{2}}\\!\\!\\!\\!\\!d\\tau_{3}\\int_{-L}^{\\tau_{3}}\\!\\!\\!\\!\\! d\\tau_{4}~ (\\tau_{1}-\\tau_{2})^{2 \\epsilon -2} (\\tau_{3}-\\tau_{4})^{2 \\epsilon -2}\\!\\!=\\nonumber\\\\\n   =\n  \n  \n   \\left(\\frac{2\\pi}{\\kappa}\\right)^{\\!\\!2}\\!\\!M^{2}\\!N\n   \\frac{\\Gamma^2\n   \\left(\\frac{1}{2}-\\epsilon \\right)}{16 \\pi^{3-2\\epsilon}}\\frac{\\sqrt{\\pi}}{ 2^{4\\epsilon}}\\frac{\\Gamma(2\\epsilon+1)}{\\Gamma(2\\epsilon+\n   \\frac{1}{2})}\\frac{(\\mu L)^{4\\epsilon}}{\\epsilon^{2}}.\n \\end{align}\n\nThe diagram \\ref{DoubleExc1class}.(b) instead leads to a different computation\n\\begin{equation}\n\\label{archetti1}\n\\begin{split}\n\\!\\!\\!\\!\n[\\ref{DoubleExc1class}.(b)]_{\\rm up}\n\\!=\\!-\\!\\left(\\frac{2\\pi}{\\kappa}\\right)^{\\!\\!2}\\!\\!M^{2}N\\frac{ \\Gamma^{2} \\left(\\frac{1}{2}-\\epsilon \\right)\n  }{16 \\pi ^{{3}-2\\epsilon} }\\frac{\\mu^{4\\epsilon}}{\\epsilon} \\frac{\\cos\\frac{\\theta}{2}}{\\cos\\frac{\\varphi}{2}}\n\\int_{0}^{L}\\!\\!\\!\\!\\!d\\tau_{1}\\!\\!\\int_{-L}^{0}\\!\\!\\!\\!\\!d\\tau_{2} (L+\\tau_{2})^{2 \\epsilon }(\\partial_{\\tau_{2}}\\!-\\partial_{\\tau_{1}})\nH(\\tau_{1},\\tau_{2}),\n\\end{split}\n\\end{equation}\nwhere $H(\\tau_{1},\\tau_{2})=(\\tau_{1}^{2}+\\tau_{2}^{2}-2\\tau_{1}\\tau_{2}\\cos\\varphi)^{-\\frac{1}{2}+\\epsilon}$. We have performed the two trivial integrations over $\\tau_{3}$ and $\\tau_{4}$ since they involve a propagator whose endpoints \nbelongs to the same edge. To extract the result we are interested in, we do not need the exact\nvalue of the remaining integral, but only its $\\epsilon-$expansion up to finite terms discussed in app. \\ref{perturbativeintegrals}. We get\n\\begin{equation}\n[\\ref{DoubleExc1class}.(b)]_{\\rm up}\\!=\\!-\\left(\\frac{2\\pi}{\\kappa}\\right)^{\\!\\!2}M^{2} N\\frac{\\cos\\frac{\\theta}{2}}{\\cos\\frac{\\varphi}{2}}\\frac{\\Gamma^{2}(\\frac{1}{2}-\\epsilon)}{16\\pi^{3-2\\epsilon}}(L\\mu)^{4\\epsilon}\\left[\\frac{1}{ \\epsilon^{2} }-\\frac{2}{\\epsilon}\\log \\left(1+\\sec \\frac{\\varphi }{2}\\right)+O(1)\\right].\n\\end{equation}\n \\begin{wrapfigure}[10]{l}{85mm}\n\\centering{\n    \\includegraphics[width=.55\\textwidth, height=.25\\textwidth]{DoubleExchange2.pdf}}\n\\vskip -.3cm\n\\caption{\\label{DoubleExc2class} Second group of double-exchange diagrams}\n\\end{wrapfigure}\nThe next step is to consider the term (B1) in \\eqref{DoubleExchange}: again we have to separate the region  of\n integration in five sub-sectors and this yields the diagrams in fig. \\ref{DoubleExc2class}.\nHowever the same reflection symmetry considered in the case of the term (A1) implies that we have just to compute \n\\ref{DoubleExc2class}.(a), \\ref{DoubleExc2class}.(b) and \\ref{DoubleExc2class}.(c).  The first one can be easily computed in closed form and it gives\n\\begin{equation}\n[\\ref{DoubleExc2class}.(a)]_{\\rm up}=\\left(\\frac{2\\pi}{\\kappa}\\right)^{\\!\\!2}\\!\\!M\\!N^{\\!2}\\frac{\\Gamma\n   \\left(\\frac{1}{2}-\\epsilon \\right)^2}{16 \\pi ^{3-2 \\epsilon}}\\frac{ (2 \\epsilon -1) }{2 (4 \\epsilon -1)}\\frac{(\\mu L)^{4\\epsilon}}{\\epsilon^{2}}.\n\\end{equation}\nConcerning the second diagram, we can trivially perform the integration over $\\tau_{2}$ and $\\tau_{3}$, obtaining\n\\begin{equation}\n\\label{archetti2}\n\\begin{split}\n[\\ref{DoubleExc2class}.(b)]_{\\rm up}=- \\left(\\frac{2\\pi}{\\kappa}\\right)^{\\!\\!2}\\!\\!N^{2} M \\frac{\\Gamma^{2}(\\frac{1}{2}-\\epsilon)}{16\\pi^{3-2\\epsilon}}\\frac{\\cos\\frac{\\theta}{2}}{\\cos\\frac{\\varphi}{2}}\\frac{\\mu^{4\\epsilon}}{\\epsilon}\n\\!\\int_{0}^{L}\\!\\!\\!\\!d\\tau_{1}\\int_{-L}^{0}\\!\\! \\!\\!d\\tau_{4}~{(-\\tau_{4})^{2 \\epsilon }}\n(\\partial_{\\tau_{4}}-\\partial_{\\tau_{1}})H(\\tau_{1},\\tau_{4}).\n\\end{split}\n\\end{equation}\nWith the help of  the results of app. \\ref{perturbativeintegrals}, we can then write the following $\\epsilon-$expansion\n\\begin{equation}\n[\\ref{DoubleExc2class}.(b)]_{\\rm up}=- \\left(\\frac{2\\pi}{\\kappa}\\right)^{\\!\\!2}\\!\\!N^{2} M \\frac{\\Gamma^{2}(\\frac{1}{2}-\\epsilon)}{16\\pi^{3-2\\epsilon}}\\frac{\\cos\\frac{\\theta}{2}}{\\cos\\frac{\\varphi}{2}}{(L\\mu)^{4\\epsilon}}\\left[\\frac{1}{2\\epsilon^{2}}+\\frac{1}{\\epsilon}\\log \\left(\\frac{1}{4} \\cos \\frac{\\varphi }{2}\\sec\n   ^4\\frac{\\varphi }{4}\\right) +O(1)\\right].\n\\end{equation}\nFor the graph \\ref{DoubleExc2class}.(c) we shall adopt a different procedure since both propagators connect different edges.  First we rearrange its integral expression as follows\n\\begin{equation}\n\\label{crossed}\n\\begin{split}\n[\\ref{DoubleExc2class}.(c)]_{\\rm up}=&-2\n\\left(\\frac{2\\pi}{\\kappa}\\right)^{\\!\\!2}\\! N^{2} M\\left(\\mu^{2\\epsilon}\\int_{0}^{L}\\!\\!\\!\\!\\!\\!d\\tau_{1}\\int_{-L}^{0}\\!\\!\\!\\!\\!\\!d\\tau_{4} S(x_{4}-x_{1})\\right)^{2}\n-\\\\\n&\\ \\  \\ \\ \\ \\  -4\\left(\\frac{2\\pi}{\\kappa}\\right)^{\\!\\!2}\\!M N^{2}  \\mu^{4\\epsilon}\\int_{0}^{L}\\!\\!\\!\\!\\!d\\tau_{1}\\int_{0}^{\\tau_{1}}\\!\\!\\!\\!\\!d\\tau_{2}\\int_{-L}^{0}\\!\\!\\!\\!\\!d\n\\tau_{3}\\int_{-L}^{\\tau_{3}}\\!\\!\\!\\!\\! d\\tau_{4}~ S(x_{2}-x_{4}) S(x_{3}-x_{1}),\n\\end{split}\n\\end{equation}\nwhere we  have separated the ``abelian\" and ``non-abelian\" part of the diagram. The former is given by the first term, which is proportional to the square of \\ref{fig:oneloopgraphs}.(e),\n while  the latter is identified \nwith the  second term in \\eqref{crossed}. This decomposition also has advantage that the leading divergence $1\/\\epsilon^{2}$ is only present in the first term.\n\nWith the help of  the results of app. \\ref{perturbativeintegrals}, we obtain  the following $\\epsilon-$expansion\nfor this diagram\n\\begin{equation}\n\\label{crossedR}\n\\begin{split}\n[\\ref{DoubleExc2class}.(c)]_{\\rm up}=&\\frac{1}{2}\\left(\\frac{2\\pi}{\\kappa}\\right)^{\\!\\!2} N^{2} M\\frac{\\Gamma^{2}(\\frac{1}{2}-\\epsilon)}{16\\pi^{3-2\\epsilon}}\n (L\\mu)^{4\\epsilon}    \\left[\\frac{\\cos\\frac{\\theta}{2}}{\\cos\\frac{\\varphi}{2}}\\left(\\frac{1}{\\epsilon}-2\\log\\left(1+\\sec\\frac{\\varphi}{2}\\right)\\right)\\right]^{2}\n-\\\\\n&\\ \\  \\ \\ \\ \\  -\\left(\\frac{2\\pi}{\\kappa}\\right)^{\\!\\!2} N^{2} M\\left(\\frac{\\cos\\frac{\\theta}{2}}{\\cos\\frac{\\varphi}{2}}\\right)^{2}\n\\frac{\\Gamma^{2}(\\frac{1}{2}-\\epsilon)}{16\\pi^{3-2\\epsilon}}\n\\frac{(L\\mu)^{4\\epsilon}}{\\epsilon}\\cos^{2}\\frac{\\varphi}{2}\\frac{\\varphi}{\\sin\\varphi}+O(1).\n\\end{split}\n\\end{equation}\n\n\\subsubsection{Vertex Diagrams} \nThe final  group of  fermionic diagrams, which are relevant for our calculation, arises when we expand in perturbation  theory  the term\n\\begin{equation}\n\\label{vertex}\n-\\frac{2\\pi i}{\\kappa}\\int_{\\Gamma}d\\tau_{\\mbox{\\tiny $\\displaystyle1\\!\\!>\\!\\! 2\\!\\!>\\!\\!3$}}\\left\\langle(\\underset{(\\mathrm{{\\bf A}_2})}{\\eta\\bar{\\psi})_1(\\psi\\bar{\\eta})_2{\\cal A}_3 }\n+\\underset{(\\mathrm{{\\bf B}_2})}{{\\cal A}_1(\\eta\\bar{\\psi})_2(\\psi\\bar{\\eta})_3}+\\underset{(\\mathrm{{\\bf C}_2})}{(\\eta\\bar{\\psi})_1\\hat{\\cal A}_2 (\\psi\\bar{\\eta})_3}\\right\\rangle,\n\\end{equation}\nappearing  in the upper block \\eqref{expaloop}. At this  order the expectation value in \\eqref{vertex} is evaluated by just considering the Wick-contractions of the monomials $({\\rm {\\bf A}_2})$, $({\\rm \\mathrm{{\\bf B}_2}})$ and $({\\rm \\mathrm{{\\bf C}_2}})$  with the tree-level gauge-fermion vertices present in the Lagrangian \\eqref{Lagra}. Then the three different contributions\ncan be rewritten as follows\n  \\begin{subequations}\n \\label{Vertex1}\n\\begin{align}\n\\!\\!\\!\\!\\! (\\mathrm{{\\bf A}_2})=& -\\left(\\frac{2\\pi}{\\kappa}\\right)^{2}N^{2}M \n ~\\int_{\\Gamma}d\\tau_{\\mbox{\\tiny $\\displaystyle1\\!\\!>\\!\\! 2\\!\\!>\\!\\!3$}}~\\eta_{1L}\\gamma_{\\nu}\\gamma^{\\mu}\\gamma_{\\lambda}\\bar\\eta_{2}^{L} \\epsilon_{\\mu\\rho\\sigma}\n\\dot{x}_{3}^{\\rho}~\n\\Gamma^{\\nu\\lambda\\sigma}(x_{1},x_{2},x_{3}),\\\\\n\\!\\!\\!\\!\\!(\\mathrm{\\mathrm{{\\bf B}_2}})=&  -\\left(\\frac{2\\pi }{\\kappa}\\right)^{2} N^{2}M~\\int_{\\Gamma}d\\tau_{\\mbox{\\tiny $\\displaystyle1\\!\\!>\\!\\! 2\\!\\!>\\!\\!3$}}~\\eta_{2L}\\gamma_{\\lambda}\\gamma^{\\mu}\\gamma_{\\nu}\\bar\\eta^{L}_{3}\n\\epsilon_{\\mu\\rho\\sigma}\\dot{x}_{1}^{\\rho}~\\Gamma^{\\sigma\\lambda\\nu}(x_{1}, x_{2}, x_{3}),\n\\\\\n\\!\\!\\!\\!\\! (\\mathrm{\\mathrm{{\\bf C}_2}}) =&- \\left(\\frac{2\\pi}{\\kappa}\\right)^{2}  N M^{2}\\int_{\\Gamma}d\\tau_{\\mbox{\\tiny $\\displaystyle1\\!\\!>\\!\\! 2\\!\\!>\\!\\!3$}}~ \\eta_{1L}\n\\gamma_{\\lambda}\\gamma^{\\mu}\\gamma_{\\nu}\n \\bar\\eta_{3}^{\\ L}\\epsilon_{\\mu\\rho\\sigma}\\dot{x}_{2}^{\\rho}\n~ \\Gamma^{\\lambda\\sigma\\nu}(x_{1},x_{2},x_{3}),\n\\end{align}\n\\end{subequations}\nwhere $\\Gamma^{\\lambda\\mu\\nu}(x_{1},x_{2}, x_{3})$\nis a short-hand notation for the three-point function in position space, defined by the integral\n\\begin{equation}\n\\label{threepoint}\n\\begin{split}\n\\Gamma^{\\lambda\\mu\\nu}(x_{1},x_{2}, x_{2})=&\\left(\\frac{\\Gamma(\\frac{1}{2}-\\epsilon)}{4\\pi^{3\/2-\\epsilon}}\\right)^{3}\\partial_{x_{1}^{\\lambda}}\\partial_{x_{2}^{\\mu}}\\partial_{x_{3}^{\\nu}}\n\\int \n\\frac{d^{3-2\\epsilon}w}{(x_{1w}^{2})^{1\/2-\\epsilon}(x_{2w}^{2})^{1\/2-\\epsilon}(x_{3w}^{2})^{1\/2-\\epsilon}}.\n\\end{split}\n\\end{equation}\n\\begin{wrapfigure}[7]{l}{89mm}\n\\vskip-.6cm\n\\centering{\n    \\includegraphics[width=.50\\textwidth, height=.25\\textwidth]{VerDiaIIIb.pdf}}\n\\vskip -.3cm\n\\caption{\\label{Vertexa} Vertex diagrams where the fermion propagators are attached to the same edge of the gluon propagator.}\n\\end{wrapfigure}\nDiagrammatically the three contributions \\eqref{Vertex1} will lead to graphs which differ for \nthe position of the  gauge field along the contour: $(\\mathrm{{\\bf A}_2})$ and $(\\mathrm{{\\bf B}_2})$ only  yield diagrams \nwhere the gluon is respectively the first or the last field we encounter when  $\\tau$ runs from \n$-L$ to $L$; $(\\mathrm{{\\bf C}_2})$  instead corresponds to diagrams where the gauge field is always located between\nthe two fermionic lines.  For instance, \n\\ref{Vertexa}.(a) and \\ref{Vertexa}.(d) originate from $(\\mathrm{{\\bf A}_2})$, \\ref{Vertexa}.(b) \nand \\ref{Vertexa}.(e) from $(\\mathrm{{\\bf C}_2})$ and \\ref{Vertexa}.($ $c) and \\ref{Vertexa}.(f) from $(\\mathrm{{\\bf B}_2})$.\n\n\nIf we now  expand the spinor bilinears in \\eqref{Vertex1} in terms of  the circuit tangent vectors  $\\dot{x}_{i}$ and  of the scalar contraction $(\\eta_{i}\\bar\\eta_{j})$, the three  contributions  $(\\mathrm{{\\bf A}_2})$, $(\\mathrm{{\\bf B}_2})$ and $(\\mathrm{{\\bf C}_2})$  can be rewritten as follows\n\\begin{subequations}\n\\label{Vertex2}\n\\begin{align}\n\\label{Vertex2a}\n(\\mathrm{{\\bf A}_2})=\n &-N^{2}M \\left(\\frac{2\\pi}{\\kappa}\\right)^{2}i(n_{1}\\cdot n_{2})\\oint_{\\tau_{1}>\\tau_{2}>\\tau_{3}}\n\\biggl[\\frac{2}{(\\eta_{2}\\bar\\eta_{1})}\n\\biggl(((\\dot{x}_{2}\\cdot\\dot{x}_{3})\\dot{x}_{1\\nu}-(\\dot{x}_{1}\\cdot\\dot{x}_{3})\\dot{x}_{2\\nu})[\\Gamma^{\\nu\\tau\\tau}+\\nonumber\\\\\n& \\phantom{\\Biggl [}+\\Gamma^{\\tau\\nu\\tau}\n-\\Gamma^{\\tau\\tau\\nu}] +\\dot{x}_{1\\sigma} \\dot{x}_{2\\nu} \\dot{x}_{3\\lambda}\\Gamma^{\\nu\\lambda\\sigma}-\\dot{x}_{1\\nu}\\dot{x}_{2\\sigma}\\dot{x}_{3\\lambda}\\Gamma^{\\nu\\lambda\\sigma}+\n\\dot{x}_{1\\sigma} \\dot{x}_{2\\lambda} \\dot{x}_{3\\nu}\\Gamma^{\\nu\\lambda\\sigma}-\\nonumber\\\\\n&-\\dot{x}_{1\\lambda}\\dot{x}_{2\\sigma}\\dot{x}_{3\\nu}\\Gamma^{\\nu\\lambda\\sigma}\\biggr)+{(\\eta_{1}\\bar\\eta_{2}) \\dot{x}_{3\\lambda}(\\Gamma^{\\tau\\lambda\\tau}-\\Gamma^{\\lambda\\tau\\tau})}\\Biggr],\\\\\n\\label{Vertex2b}\n(\\mathrm{{\\bf B}_2})= &  -N^{2}M\\left(\\frac{2\\pi }{\\kappa}\\right)^{2} i(n_{3}\\cdot n_{2})\\oint_{\\tau_{1}>\\tau_{2}>\\tau_{3}}~\\Biggl[\\frac{2}{(\\eta_{3}\\bar\\eta_{2})}\n\\biggl([(\\dot{x}_{1}\\cdot\\dot{x}_{3})\\dot{x}_{2\\nu}-(\\dot{x}_{2}\\cdot\\dot{x}_{1})x_{3\\nu}][\\Gamma^{\\tau\\tau\\nu}+\\nonumber\\\\\n& \\phantom{\\Biggl [}+\\Gamma^{\\tau\\nu\\tau}-\\Gamma^{\\nu\\tau\\tau}]+\\dot{x}_{1\\lambda}\\dot{x}_{2\\nu} \\dot{x}_{3\\sigma} \\Gamma^{\\nu\\lambda\\sigma}-\\dot{x}_{1\\lambda}\\dot{x}_{2\\sigma}\\dot{x}_{3\\nu}\\Gamma^{\\nu\\lambda\\sigma}+\\dot{x}_{1\\sigma}\\dot{x}_{2\\nu} \\dot{x}_{3\\lambda} \\Gamma^{\\nu\\lambda\\sigma}-\\nonumber\\\\\n&-\\dot{x}_{1\\sigma}\\dot{x}_{2\\lambda}\\dot{x}_{3\\nu}\\Gamma^{\\nu\\lambda\\sigma}\\biggr)+{(\\eta_{2}\\bar\\eta_{3})\\dot{x}_{1\\nu}(\\Gamma^{\\tau\\tau\\nu}-\\Gamma^{\\tau\\nu\\tau})}\\Biggr],\\displaybreak[2]\n\\\\\n\\label{Vertex2c}\n(\\mathrm{{\\bf C}_2})=&-N M^{2} \\left(\\frac{2\\pi}{\\kappa}\\right)^{2}  i(n_{1}\\cdot n_{3})\\oint_{\\tau_{1}>\\tau_{2}>\\tau_{3}}\\Biggl[\\frac{2}{(\\eta_{3}\\bar\\eta_{1})}\n\\biggl(((\\dot{x}_{2}\\cdot\\dot{x}_{3})\\dot{x}_{1\\nu}-(\\dot{x}_{1}\\cdot\\dot{x}_{2})\\dot x_{3\\nu})[\\Gamma^{\\tau\\tau\\nu}+\\nonumber\\\\\n& \\phantom{\\Biggl [}+\\Gamma^{\\nu\\tau\\tau}-\\Gamma^{\\tau\\nu\\tau}]+\\dot{x}_{1\\sigma} \\dot{x}_{3\\nu} \\dot{x}_{2\\lambda}\\Gamma^{\\lambda\\sigma\\nu}-\\dot{x}_{1\\nu}\\dot{x}_{3\\sigma}\\dot{x}_{2\\lambda}\\Gamma^{\\lambda\\sigma\\nu}+\\dot{x}_{1\\sigma} \\dot{x}_{3\\lambda} \\dot{x}_{2\\nu}\\Gamma^{\\lambda\\sigma\\nu}-\\nonumber\\\\\n&-\\dot{x}_{1\\lambda}\\dot{x}_{3\\sigma}\\dot{x}_{2\\nu}\\Gamma^{\\lambda\\sigma\\nu}\\biggr)+(\\eta_{1}\\bar\\eta_{3})\\dot{x}_{2\\nu}(\\Gamma^{\\tau\\tau\\nu}-\\Gamma^{\\nu\\tau\\tau})\\Biggr],\n\\end{align}\n\\end{subequations}\nwhere we have dropped all the terms which vanish for planar contours.  To begin with, we   \n shall consider \n the family of diagrams  of fig.~\\ref{Vertexa}, whe\\-re all the bosonic and fermionic lines terminate\non the same edge of the  cusp. In this case all the  terms    proportional to the factor\n$2\/(\\eta_{i}\\bar\\eta_{j})$ in \\eqref{Vertex2} drop out  because the tangent vectors obey the relation\n\\begin{equation}\n\\label{eq1}\n\\dot{x}_{1}=\\dot{x}_{2}=\\dot{x}_{3},\n\\end{equation}\nfor each diagram in fig.~\\ref{Vertexa}. Only the last  terms in  \\eqref{Vertex2a}, \\eqref{Vertex2b} and  \\eqref{Vertex2c}    that  are proportional to the bilinear $(\\eta_{i}\\bar\\eta_{j})$ are  different from zero and we are left with\n\\begin{subequations}\n\\label{Vertex3}\n\\begin{align}\n\\label{Vertex3a}\n(\\mathrm{{\\bf A}_2})=\n &2 N^{2}M \\left(\\frac{2\\pi}{\\kappa}\\right)^{2}\\oint_{\\tau_{1}>\\tau_{2}>\\tau_{3}}\n\\dot{x}_{3\\lambda}(\\Gamma^{\\tau\\lambda\\tau}-\\Gamma^{\\lambda\\tau\\tau})\\equiv-\\left(\\frac{2\\pi}{\\kappa}\\right)^{2} N^{2}M (\\mathfrak{a})\\\\\n(\\mathrm{{\\bf B}_2})= &  2 N^{2}M\\left(\\frac{2\\pi }{\\kappa}\\right)^{2} \\oint_{\\tau_{1}>\\tau_{2}>\\tau_{3}}~\\dot{x}_{1\\nu}(\\Gamma^{\\tau\\tau\\nu}-\\Gamma^{\\tau\\nu\\tau})\\equiv-\\left(\\frac{2\\pi}{\\kappa}\\right)^{2} N^{2}M (\\mathfrak{b})\\displaybreak[2]\n\\\\\n(\\mathrm{{\\bf C}_2})=&2N M^{2} \\left(\\frac{2\\pi}{\\kappa}\\right)^{2} \\oint_{\\tau_{1}>\\tau_{2}>\\tau_{3}}\\dot{x}_{2\\nu}(\\Gamma^{\\tau\\tau\\nu}-\\Gamma^{\\nu\\tau\\tau})\\equiv-\\left(\\frac{2\\pi}{\\kappa}\\right)^{2} N M^{2} (\\mathfrak{c}),\n\\end{align}\n\\end{subequations}\nwhere we used that $\\eta_{i}\\bar\\eta_{j}=2i$ and $(n_{i}\\cdot n_{j})=1$. There is a further simplification: in fact  we do not have to compute  all  the diagrams originating from $(\\mathrm{{\\bf A}_2})$,\n$(\\mathrm{{\\bf B}_2})$ and $(\\mathrm{{\\bf C}_2})$ and depicted in fig. \\ref{Vertexa}. First of all, we can  restrict ourselves to considering only the   diagrams \\ref{Vertexa}.(a), \\ref{Vertexa}.(b) and \\ref{Vertexa}.($ $c).  The other three graphs will simply double the final result. Next, we note that the following identity holds for this subclass of diagrams\n\\begin{equation}\n(\\mathfrak{a})+(\\mathfrak{b})=(\\mathfrak{c}),\n\\end{equation}\n{\\it i.e.} it is sufficient to evaluate only the integral  \n\\begin{align}\n\\label{t8}\n(\\mathfrak{c})=&-2 \\int_{-L}^{0}\\!\\!\\!\\!d\\tau_{1}\\int_{-L}^{\\tau_{1}}\\!\\!\\!\\!d\\tau_{2}\\int_{-L}^{\\tau_{2}}\\!\\!\\!\\!d\\tau_{3}~{\\dot{x}_{2\\nu}(\\Gamma^{\\tau\\tau\\nu}-\\Gamma^{\\nu\\tau\\tau})}\n\\end{align}\n to reconstruct the result of all the diagrams in fig. \\ref{Vertexa}.  Moreover the three-point functions appearing\n in \\eqref{t8}  always possess two contracted indices: in this case the integral \\eqref{threepoint} can be easily evaluated in terms of product of scalar propagators and one finds\n \\begin{equation}\n \\label{525}\n \\Gamma^{\\tau\\tau\\nu}=\\partial_{x_{3}^{\\nu}}\\Phi_{3,12},\\ \\ \\ \\ \n\\Gamma^{\\tau\\nu\\tau}\n=\n\\partial_{x_{2}^{\\nu}}\\Phi_{2,13},\\ \\ \\ \\ \n\\Gamma^{\\nu\\tau\\tau}\n=\\partial_{x_{1}^{\\nu}}\\Phi_{1,23},\n \\end{equation}\n where \n \\begin{equation}\n\\begin{split}\n\\Phi_{i,jk}=& -\\frac{\\Gamma^{2}(1\/2-\\epsilon)}{32\\pi^{3-2\\epsilon}}\\!\\!\n\\left[\\frac{1}{(x^{2}_{ij})^{\\frac{1}{2}-\\epsilon}(x^{2}_{ik})^{\\frac{1}{2}-\\epsilon}}-\\frac{1}{(x^{2}_{ij})^{\\frac{1}{2}-\\epsilon}(x^{2}_{kj})^{\\frac{1}{2}-\\epsilon}}-\\frac{1}{(x^{2}_{ik})^{\\frac{1}{2}-\\epsilon}(x^{2}_{jk})^{\\frac{1}{2}-\\epsilon}}\\right]\\!\\!.\n\\end{split}\n \\end{equation}\nSee  appendix $C$  for more details. With the help of this result, and recalling \\eqref{eq1}, we can \nshow that the integrand in \\eqref{t8} only contains total derivatives and can be easily computed\n\\begin{align}\n(\\mathfrak{c})=&-2 \\int_{-L}^{0}\\!\\!\\!\\!d\\tau_{1}\\int_{-L}^{\\tau_{1}}\\!\\!\\!\\!d\\tau_{2}\\int_{-L}^{\\tau_{2}}\\!\\!\\!\\!d\\tau_{3}~\\left(\\frac{d}{d\\tau_{3}}\\Phi_{3,12}-\\frac{d}{d\\tau_{1}}\\Phi_{1,23}\\right)\n=\\nonumber\\\\\n=&-2 \\int_{-L}^{0} \\!\\!\\!\\! d\\tau_{1}\\int_{-L}^{\\tau_{1}}\\!\\!\\!\\! d\\tau_{2}(\\Phi_{2,12}+\\Phi_{1,12}-\\Phi_{-L,12}-\\Phi_{0,12})=\\nonumber\\\\\n=& 2  \\frac{\\Gamma^{2}(1\/2-\\epsilon)}{32\\pi^{3-2\\epsilon}} L^{4 \\epsilon } \\left(\\frac{1}{2 \\epsilon ^2}+\\frac{1}{2 \\epsilon }+O\\left(1\\right)\\right).\n\\end{align}\n\\begin{wrapfigure}[8]{l}{85mm}\n\\centering{\n    \\includegraphics[width=.45\\textwidth, height=.17\\textwidth]{VerDiaIVb.pdf}}\n\\vskip -.3cm\n\\caption{\\label{Vertexb} Vertex diagrams with fermionic propagators attached to the opposite edge of the gluon propagator.}\n\\end{wrapfigure}\nNext we consider  the case where the fermions are both attached to the same line, but the gluon is not. \nWe have the two possibilities depicted in fig. \\ref{Vertexb}. The diagram \\ref{Vertexb}.(a) originates from the contribution\n\\eqref{Vertex2a} when considering the region of integration $-L\\le\\tau_{3}\\le 0$ and $0\\le \\tau_{2}\\le\n\\tau_{1}\\le L$. \n\nThe diagram \\ref{Vertexb}.(b) is instead obtained from \\eqref{Vertex2b}, when   $-L\\le \\tau_{3}\\le\\tau_{2}\\le 0$, $0\\le\\tau_{1}\\le L$ and $\\dot{x}_{3}=\\dot{x}_{2}$.  No contribution of this kind is instead contained in \\eqref{Vertex2c}. \n Since the two graphs in fig. \\ref{Vertexb}  are related by a reflection with respect to the axis bisecting the cusp, they yield the same result and thus we have to  compute only one of them, {\\it e.g. } \\ref{Vertexb}.(a).\n For this diagram all the terms in \\eqref{Vertex2a}, which are not proportional to $(\\eta_{1}\\bar\\eta_{2})$,  will vanish when we use that  $\\dot{x}_{1}=\\dot{x}_{2}$  and so we get an expression that is similar  to the one considered in \n\\eqref{Vertex3a}:\n \\begin{equation}\n[\\ref{Vertexb}.(a)]_{\\rm up}=2 N^{2}M \\left(\\frac{2\\pi}{\\kappa}\\right)^{2}\\int_{0}^{L}\\!\\!\\!\\! d\\tau_{1}\\int_{0}^{\\tau_{1}}\\!\\!\\!\\!d\\tau_{2}\\int_{-L}^{0}\\!\\!\\!\\!d\\tau_{3}~\\dot{x}_{3\\lambda}(\\Gamma^{\\tau\\lambda\\tau}-\\Gamma^{\\lambda\\tau\\tau}).\n\\end{equation}\nIn order to compute this integral we first observe that the integrand can be rearranged as follows\n\\begin{equation}\n\\label{cache}\n\\begin{split}\n &\\dot x_{3\\lambda}\n( \\Gamma^{\\tau\\lambda\\tau}-\\Gamma^{\\lambda\\tau\\tau})=\\dot x_{3}\\cdot\\partial_{x_{2}}\\Phi_{2,13}-\\dot x_{3}\\cdot\\partial_{x_{1}}\\Phi_{1,23}=\\dot x_{3}\\cdot\\partial_{x_{2}}(\\Phi_{2,13}+\\Phi_{1,23})+\\frac{d}{d\\tau_{3}}\\Phi_{1,23}\\\\\n&=-\\left(\\frac{\\Gamma(1\/2-\\epsilon)}{4\\pi^{3\/2-\\epsilon}}\\right)^{2}\n\\frac{1}{(x^{2}_{13})^{1\/2-\\epsilon}}\\frac{d}{d \\tau_{3}}\n\\frac{1}{(x^{2}_{23})^{1\/2-\\epsilon}}+\\frac{d}{d\\tau_{3}}\\Phi_{1,23}.\n\\end{split}\n\\end{equation}\nWe have two separate contributions, which both appear in the list considered in appendix \\ref{perturbativeintegrals}  (see eqs. \\eqref{D1} and \\eqref{D2}) and thus we can immediately write the final result\n\\begin{equation}\n\\!\\!\\!\n[\\ref{Vertexb}.(a)]_{\\rm up}\\!=\n-N^{2}M \\left(\\frac{2\\pi}{\\kappa}\\right)^{2}\n  L^{4\\epsilon}\\left(\\frac{\\Gamma(1\/2-\\epsilon)}{4\\pi^{3\/2-\\epsilon}}\\right)^{2}\\biggl[\\frac{1}{\\epsilon} \\left({\\log \\left(\\cos \\frac{\\varphi }{2}\\right)}-\\frac{1}{2 }\\varphi  \\cot \\varphi \\right)\\!+O(1)\\biggr].\n\\end{equation}\nThe final set of diagrams that we have to consider are those where the two fermions end on  different edges of \nthe cusp. We have four possible graphs  of this kind, which simply\n   \\begin{wrapfigure}[12]{l}{85mm}\n\\centering{\n    \\includegraphics[width=.40\\textwidth, height=.28\\textwidth]{VerDiaV.pdf}}\n\\vskip -.3cm\n\\caption{\\label{Vertexc} Vertex diagrams with fermionic propagators are attached to opposite edges.}\n\\end{wrapfigure}\n differ for the position of the gluon line, and they are displayed in fig.  \\ref{Vertexc}. The diagrams  \\ref{Vertexc}.(a) and  \\ref{Vertexc}.(d) are obtained respectively from \\eqref{Vertex2a} and \\eqref{Vertex2b}\n when considering the region of integrations  ${\\rm (I)}=\\{-L\\le\\tau_{3}\\le\\tau_{2}\\le0$ and $0\\le\\tau_{1}\\le L\\}$  and  ${\\rm (II)}=\\{-L\\le\\tau_{3}\\le 0$ and $0\\le\\tau_{2}\\le\\tau_{1}\\le L\\}$. The diagrams \\ref{Vertexc}.(b) and  \\ref{Vertexc}.(c) originate instead from \\eqref{Vertex2c} when choosing either the  range  (I) or (II) for the parameters $\\tau_{i}$. Again graphs, which are related by a reflection with respect the axis bisecting the cusp, produce the same result and we focus our attention only on \\ref{Vertexc}.(a)  and  \\ref{Vertexc}.(b).\n\nTo begin with, we shall  factor out from both diagrams the color and $R-$symmetry dependence  and we shall write\n\\begin{equation}\n [\\ref{Vertexc}.(a)]_{{\\rm up}}\\equiv -N^{2}M \\left(\\frac{2\\pi}{\\kappa}\\right)^{2}\\cos\\frac{\\theta}{2}~ \\mathcal{I}_{(a)}\\ \\ \\ \\ \\  [\\ref{Vertexc}.(b)]_{{\\rm up}}\\equiv -N M^{2} \\left(\\frac{2\\pi}{\\kappa}\\right)^{2}  \\cos\\frac{\\theta}{2} ~\\mathcal{I}_{(b)}.\n \\end{equation}\nIn order to simplify our analysis we shall construct the two independent combinations $\\mathcal{I}_{(a)}-\\mathcal{I}_{(b)}$ and $\\mathcal{I}_{(a)}+\\mathcal{I}_{(b)}$. The former is the only combination of the two integrals appearing in the final result when we would take the super-trace  of  the Wilson-loop and it is given by\n\\begin{align}\n\\label{supertrace}\n\\mathcal{I}_{(a)}-\\mathcal{I}_{(b)}\\!=&\\frac{2 i}{(\\eta_{2}\\bar\\eta_{1})}\\int_{0}^{L}\\!\\!\\!\\!\\!d\\tau_{1}\\!\\!\\int_{-L}^{0}\\!\\!\\!\\!\\!d\\tau_{2}\\!\\!\\int_{-L}^{\\tau_{2}}\\!\\!\\!\\!\\!d\\tau_{3}\\!\n\\left(\\!\n\\dot{x}_{1\\nu}-(\\dot{x}_{1}\\cdot\\dot{x}_{3})\\dot{x}_{2\\nu}-\\frac{(\\eta_{1}\\bar\\eta_{2}) (\\eta_{2}\\bar\\eta_{1})}{2}\n\\dot{x}_{3\\nu}\\!\\right)\\!\\left(\\Gamma^{\\tau\\tau\\nu}-\\Gamma^{\\tau\\nu\\tau}\\right)\\!=\\nonumber\\\\\n=&\\frac{2 i}{(\\eta_{2}\\bar\\eta_{1})}\\int_{0}^{L}\\!\\!\\!\\!d\\tau_{1}\\int_{-L}^{0}\\!\\!\\!\\!d\\tau_{2}\\int_{-L}^{\\tau_{2}}\\!\\!\\!\\!d\\tau_{3}\n\\left(\\dot{x}_{1\\nu}+\n\\dot{x}_{2\\nu}\\right)\\left(\\Gamma^{\\tau\\tau\\nu}-\\Gamma^{\\tau\\nu\\tau}\\right).\n\\end{align}\nIn \\eqref{supertrace} we were able to get rid of all the terms containing a three-point function contracted with  three $\\dot{x}_{i}$, thanks to the identity \\eqref{C15a} and to the equality $\\dot{x}_{2}=\\dot{x}_{3}$, which holds for these diagrams.  We can now use the  relations \\eqref{525} and the invariance under translation of the function $\\Phi_{i,jk}$ to rewrite the integrand as follows\n\\begin{equation}\n\\label{5.33}\n\\begin{split}\n\\left(\n\\dot{x}_{1\\nu}+\\dot x_{2\\nu}\\right)\\left(\\Gamma^{\\tau\\tau\\nu}-\\Gamma^{\\tau\\nu\\tau}\\right)\n=&\\frac{d}{d \\tau_{3}} \\Phi_{3,12}-\\frac{d}{d \\tau_{2}} \\Phi_{2,13}-\\frac{d}{d \\tau_{1}} \\Phi_{3,12}+\\\\\n&+\\left(\\frac{\\Gamma(1\/2-\\epsilon)}{4\\pi^{3\/2-\\epsilon}}\\right)^{2}\n\\frac{1}{(x^{2}_{13})^{1\/2-\\epsilon}}\\frac{d}{d \\tau_{1}}\n\\frac{1}{(x^{2}_{12})^{1\/2-\\epsilon}}.\n\\end{split}\n\\end{equation}\nThe integration over the circuit  can be performed by means of the results given in app. \\ref{perturbativeintegrals} and we find\n\\begin{align}\n\\label{supertracea}\n\\mathcal{I}_{(a)}-\\mathcal{I}_{(b)}\\!=&\\frac{L^{4\\epsilon}}{2\\cos\\frac{\\varphi}{2}}\n\\left(\\frac{\\Gamma(1\/2-\\epsilon)}{4\\pi^{3\/2-\\epsilon}}\\right)^{2}\n\\left[\\frac{1}{2\\epsilon}\\frac{\\varphi}{\\sin\\varphi}+\\frac{1}{\\epsilon}\\log\\left(\\cos\\frac{\\varphi}{2}\\right)+\\frac{1}{4\\epsilon^{2}}-\\right.\\\\\n&\\left.-\\frac{1}{\\epsilon} \\left(\\frac{\\varphi }{2 } \\cot \\varphi -{\\log \\left(\\cos \\frac{\\varphi }{2}\\right)}\\right)+\\frac{1}{4\\epsilon^{2}}+O(1)\\right]=\\displaybreak[2]\\\\\n=&\\left(\\frac{\\Gamma(1\/2-\\epsilon)}{4\\pi^{3\/2-\\epsilon}}\\right)^{2}\\frac{{L^{4\\epsilon}}}{4\\epsilon}\\frac{\\varphi}{\\sin\\frac{\\varphi}{2}}+O(1).\n\\end{align}\nThe sum  $\\mathcal{I}_{(a)}+\\mathcal{I}_{(b)}$ is instead the only  combination appearing in the final  result if we  take the trace of the loop operator.  Its expression is less  elegant than the one for the difference and it is given by\n\\begin{align}\n\\label{trace}\n&\\mathcal{I}_{(a)}+\\mathcal{I}_{(b)}=\ni\\int_{0}^{L}\\!\\!\\!\\!\\!d\\tau_{1}\\!\\!\\int_{-L}^{0}\\!\\!\\!\\!\\!d\\tau_{2}\\!\\!\\int_{-L}^{\\tau_{2}}\\!\\!\\!\\!\\!d\\tau_{3}~\\Biggl[\n\\frac{2}{(\\eta_{2}\\bar\\eta_{1})}\n\\biggl(2(\\dot{x}_{1\\nu}+\\dot x_{3\\nu})\\Gamma^{\\nu\\tau\\tau}\n+\\dot{x}_{2\\nu} \\dot{x}_{1\\lambda} \\dot{x}_{3\\sigma}\\Gamma^{\\nu\\lambda\\sigma}+\\nonumber\\\\\n&+\\dot{x}_{2\\lambda}\\dot{x}_{1\\sigma}\\dot{x}_{3\\nu}\\Gamma^{\\nu\\lambda\\sigma}-2\n\\dot{x}_{1\\nu} \\dot{x}_{3\\lambda} \\dot{x}_{2\\sigma}\\Gamma^{\\nu\\lambda\\sigma}\\biggr)+\n(\\eta_{1}\\bar\\eta_{2}) ~\\dot{x}_{3\\nu}(\\Gamma^{\\tau\\nu\\tau}+\\Gamma^{\\tau\\tau\\nu})\\Biggr].\n\\end{align}\nIt is not difficult to realize that the integrand in \\eqref{trace} is symmetric when exchanging $\\tau_{2}$\nwith $\\tau_{3}$: this allows us to extend the integration over $\\tau_{3}$ up to $0$ provided dividing the result by two. We can reorganize \\eqref{trace} as follows\n\\begin{align}\n\\label{trace1}\n\\mathcal{I}_{(a)}+\\mathcal{I}_{(b)}=&\n\\frac{i}{2}\\int_{0}^{L}\\!\\!\\!\\!\\!d\\tau_{1}\\!\\!\\int_{-L}^{0}\\!\\!\\!\\!\\!d\\tau_{2}\\!\\!\\int_{-L}^{0}\\!\\!\\!\\!\\!d\\tau_{3}~\\Biggl[\n\\frac{2}{(\\eta_{2}\\bar\\eta_{1})}\n\\biggl(2(\\dot{x}_{1\\nu}+\\dot x_{3\\nu})\\Gamma^{\\nu\\tau\\tau}\n+\\dot{x}_{2\\nu} \\dot{x}_{1\\lambda} \\dot{x}_{3\\sigma}\\Gamma^{\\nu\\lambda\\sigma}+\\nonumber\\\\\n&+\\dot{x}_{2\\lambda}\\dot{x}_{1\\sigma}\\dot{x}_{3\\nu}\\Gamma^{\\nu\\lambda\\sigma}-2\n\\dot{x}_{1\\nu} \\dot{x}_{3\\lambda} \\dot{x}_{2\\sigma}\\Gamma^{\\nu\\lambda\\sigma}\\biggr)+\n(\\eta_{1}\\bar\\eta_{2}) ~\\dot{x}_{3\\nu}(\\Gamma^{\\tau\\nu\\tau}+\\Gamma^{\\tau\\tau\\nu})\\Biggr]=\\nonumber\\\\\n=&\n{i}\\int_{0}^{L}\\!\\!\\!\\!\\!d\\tau_{1}\\!\\!\\int_{-L}^{0}\\!\\!\\!\\!\\!d\\tau_{2}\\!\\!\\int_{-L}^{0}\\!\\!\\!\\!\\!d\\tau_{3}~\\Biggl[\n\\frac{2}{(\\eta_{2}\\bar\\eta_{1})}\n\\biggl((\\dot{x}_{1\\nu}+\\dot x_{3\\nu})\\Gamma^{\\nu\\tau\\tau}\n+\\dot{x}_{2\\nu} \\dot{x}_{1\\lambda} \\dot{x}_{3\\sigma}\\Gamma^{\\nu\\lambda\\sigma}-\\nonumber\\\\\n&-\\dot{x}_{1\\nu} \\dot{x}_{3\\lambda} \\dot{x}_{2\\sigma}\\Gamma^{\\nu\\lambda\\sigma}\\biggr)+\n(\\eta_{1}\\bar\\eta_{2}) ~\\dot{x}_{3\\nu}\\Gamma^{\\tau\\tau\\nu}\\Biggr].\n\\end{align}\nIn the second equality in \\eqref{trace1} we have identified all the terms which differ by a permutation of\n$\\tau_{2}$ with $\\tau_{3}$, being trivially equivalent. We can distinguish two types of contributions:\none containing only contracted three-point functions and the other  where the three-point functions are saturated with \nthree $\\dot x_{i}$. The former can be rewritten in terms of the function $\\Phi_{i,jk}$ by means  of the relations \\eqref{525}\nand we obtain\n\\begin{align}\n\\label{5.39}\n&\n{i}\\int_{0}^{L}\\!\\!\\!\\!\\!d\\tau_{1}\\!\\!\\int_{-L}^{0}\\!\\!\\!\\!\\!d\\tau_{2}\\!\\!\\int_{-L}^{0}\\!\\!\\!\\!\\!d\\tau_{3}~\\Biggl[\n\\frac{2}{(\\eta_{2}\\bar\\eta_{1})}\n(\\dot{x}_{1\\nu}+\\dot x_{3\\nu})\\Gamma^{\\nu\\tau\\tau}\n+\n(\\eta_{1}\\bar\\eta_{2}) ~\\dot{x}_{3\\nu}\\Gamma^{\\tau\\tau\\nu}\\Biggr]=\\nonumber\\\\\n=&{i}\\int_{0}^{L}\\!\\!\\!\\!\\!d\\tau_{1}\\!\\!\\int_{-L}^{0}\\!\\!\\!\\!\\!d\\tau_{2}\\!\\!\\int_{-L}^{0}\\!\\!\\!\\!\\!d\\tau_{3}~\\Biggl[\n\\frac{2}{(\\eta_{2}\\bar\\eta_{1})}\\biggl(\n\\frac{d}{d\\tau_{1}} \\Phi_{1,23}-\\frac{d}{d\\tau_{2}} \\Phi_{1,23}-\n\\frac{d}{d\\tau_{3}}\\Phi_{1,23}\\biggr)+(\\eta_{1}\\bar\\eta_{2})\\frac{d}{d\\tau_{3}} \\Phi_{3,12}\\Biggr].\n\\end{align}\nThe divergent part of these integrals can be extracted from the table of integrals presented  in app.  \\ref{perturbativeintegrals} and we find\n\\begin{equation}\n\\label{trace3}\n\\frac{\\Gamma^{2}(1\/2-\\epsilon)}{16\\pi^{3-2\\epsilon}}\\left[\\frac{L^{4\\epsilon}}{\\cos\\frac{\\varphi}{2}}\\left[\\frac{2}{\\epsilon}\\log \\left(\\sec \\left(\\frac{\\varphi }{2}\\right)+1\\right)-\\frac{1}{2\\epsilon^{2}}\\right]+\\frac{L^{4\\epsilon}}{\\epsilon}\\cos\\frac{\\varphi}{2}\\log\\left(\\cos\\frac{\\varphi}{2}\\right)+O(1)\\right].\n\\end{equation}\nThe procedure for determining the divergences of the latter contribution in \\eqref{trace1} is more delicate, since we have to deal with the untraced three-point function.\nAfter a careful analysis, one gets\n\\begin{align}\n\\label{trace4}\n\\frac{2i}{(\\eta_{2}\\bar\\eta_{1})}\\int_{0}^{L}\\!\\!\\!\\!\\!d\\tau_{1}\\!\\!\\int_{-L}^{0}\\!\\!\\!\\!\\!d\\tau_{2}\\!\\!\\int_{-L}^{0}\\!\\!\\!\\!\\!d\\tau_{3}~\n\\biggl(&\n\\dot{x}_{2\\nu} \\dot{x}_{1\\lambda} \\dot{x}_{3\\sigma}\\Gamma^{\\nu\\lambda\\sigma}-\\dot{x}_{1\\nu} \\dot{x}_{3\\lambda} \\dot{x}_{2\\sigma}\\Gamma^{\\nu\\lambda\\sigma}\\biggr)=\\nonumber\\\\\n&=-\\frac{\\Gamma^{2}(1\/2-\\epsilon)}{16\\pi^{3-2\\epsilon}}\\frac{{L^{4\\epsilon}}}{\\epsilon}\\cos \\frac{\\varphi }{2}\\log \\left(\\cos \\frac{\\varphi }{2}\\right)+O(1).\n\\end{align}\nIf we sum \\eqref{trace3} and \\eqref{trace4}, we can  finally write down the result for $\\mathcal{I}_{(a)}+\\mathcal{I}_{(b)}$\n\\begin{equation}\n\\mathcal{I}_{(a)}+\\mathcal{I}_{(b)}=\\frac{\\Gamma^{2}(1\/2-\\epsilon)}{16\\pi^{3-2\\epsilon}}\\frac{L^{4\\epsilon}}{\\cos\\frac{\\varphi}{2}}\\left[\\frac{2}{\\epsilon}\\log \\left(\\sec \\left(\\frac{\\varphi }{2}\\right)+1\\right)-\\frac{1}{2\\epsilon^{2}}\\right]+O(1).\n\\end{equation}\nThis completes the evaluation of the divergent part of all diagrams at two loops.\n\\section{The final result: summing and renormalizing}\nIn this section we shall add up the different diagrams which appear at two loops.  Because we are actually working with an open contour, we have in principle two possibilities  to perform this  sum: we can take the trace [$\\mathcal{W}_{+}$ in \\eqref{Wpm}] or the super-trace [$\\mathcal{W}_{-}$ in \\eqref{Wpm}].  As we shall see, the first choice, that is the correct one for closed contours, appears to be consistent with an exponentiated form. We also discuss the renormalization of our result, paying particular attention to the peculiarities arising in three dimensions and in the presence of the exotic fermionic couplings.\n\n\\subsection{Taking the trace}\nLet us consider the case of the trace. The bosonic bubble diagrams yield  a  four-dimensional-like  contribution given by\n\\begin{equation}\n\\mathds{B}=-g(\\epsilon)\\left(\\cos\\varphi-\\cos^{2}\\frac{\\theta}{2}\\right )\\frac{1}{\\epsilon}\\frac{\\varphi}{\\sin\\varphi},\n\\end{equation}\nwhere we have introduced the short-hand notation \n\\begin{equation}\ng(\\epsilon)= M N\\left(\\frac{2\\pi}{\\kappa}\\right)^{\\!\\!2}\\!\\frac{\\Gamma^{2}\\!\\left(\\frac{1}{2}-\\epsilon\\right)}{16 \\pi^{3-2\\epsilon}}(\\mu L)^{4\\epsilon}\n\\end{equation}\n for future convenience.\nThe fermionic bubble instead cancels when we take the trace, since it is odd in the exchange $N\\leftrightarrow M$.  The total  result for the complete set of \ndouble-exchange diagrams is more elaborate and it can be usefully cast in the form\n\\begin{equation}\n\\begin{split}\n\\mathds{D}=&2 g(\\epsilon)\n\\Biggl[\\frac{1}{\\epsilon ^2}\\left[2-\\frac{3}{2} \\frac{\\cos \\frac{\\theta }{2}}{ \\cos \\frac{\\varphi }{2}}\\right]+\\frac{1}{\\epsilon }\\left[ \\frac{\\cos \\frac{\\theta }{2}}{ \\cos \\frac{\\varphi }{2}}\\left(4 \\log \\left(\\sec \\frac{\\varphi }{2}+1\\right)+\\log \\left(\\cos \\frac{\\varphi\n   }{2}\\right)\\right)+1\\right]+\\\\\n   &+\\frac{1}{4}   \\left[\\frac{\\cos\\frac{\\theta}{2}}{\\cos\\frac{\\varphi}{2}}\\left(\\frac{1}{\\epsilon}-2\\log\\left(1+\\sec\\frac{\\varphi}{2}\\right)\\right)\\right]^{2}\n -\\frac{1}{2\\epsilon}\\cos^{2}\\frac{\\theta}{2}\n\\frac{\\varphi}{\\sin\\varphi}+O(1)\\Biggr].\n\\end{split}\n\\end{equation}\nThe  diagrams which contain the gauge-fermion interaction  yield  instead the following result\n\\begin{equation}\n\\mathds{V}\\!=\\!\n\\frac{g(\\epsilon)}{\\epsilon ^2}\\!\\left[{\\frac{\\cos\\frac{\\theta}{2}}{\\cos\\frac{\\varphi}{2}}-2}\\right]+\\frac{g(\\epsilon)}{\\epsilon}\\!\\left[{\\varphi  \\cot \\varphi -2\\! \\left(\\!2 \\frac{\\cos\\frac{\\theta}{2}}{\\cos\\frac{\\varphi}{2}}\\log \\left(\\sec \\frac{\\varphi }{2}+1\\right)+\\log \\left(\\cos\n   \\frac{\\varphi }{2}\\right)\\!+\\!1\\!\\right)}\\right]\\!+\\!O\\left(1\\right)\\!.\n\\end{equation}\nWe shall now sum these three contributions in order to obtain the unrenormalized value of $\\mathcal{W}_{+}$ in \\eqref{Wpm} at two loops\n\\begin{equation}\n\\begin{split}\n[\\mathcal{W}_{+}]_{\\rm 2-loop}=\n&\\frac{g(\\epsilon)}{2\\epsilon^{2}}\\left(\\frac{\\cos\\frac{\\theta}{2}}{\\cos\\frac{\\varphi}{2}}-2\\right)^{2}+\\frac{2g(\\epsilon)}{\\epsilon}\\frac{\\cos\\frac{\\theta}{2}}{\\cos\\frac{\\varphi}{2}}\\left(2-\\frac{\\cos\\frac{\\theta}{2}}{\\cos\\frac{\\varphi}{2}}\\right)\\log\\left(\\sec \\frac{\\varphi }{2}+1\\right)+\\\\\n&\\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ +\\frac{2g(\\epsilon)}{\\epsilon}\\log\\left(\\cos\\frac{\\varphi}{2}\\right)\\left(\\frac{\\cos\\frac{\\theta}{2}}{\\cos\\frac{\\varphi}{2}}-1\\right)+O(1).\n\\end{split}\n\\end{equation} \nIn this expression  the structure of the  generalized potential is not manifest. Crucially we observe the presence of double-poles that are not expected to appear in the final expression of the generalized potential. In conventional Wilson loops, where only bosonic couplings are concerned, double-poles at two loops are simply understood as coming from the square of the one-loop result, by virtue of the non-abelian exponentiation theorem \\cite{Exp} (that holds even at renormalized level). The non-trivial contribution at second order in perturbation theory comes from the so-called maximally non-abelian part and in ${\\cal N}=4$ SYM, for example, involves crossed non-planar bosonic exchanges and interacting diagrams, stretching between the two lines. In our case, due to the presence of the fermionic couplings, we do not have an \nestablished  exponentiation theorem at hand and we were forced to compute the full two-loop  contribution to the quantum average.  Incidentally, for our loops, double-poles appear both from exchange and interacting diagrams at variance with ${\\cal N}=4$ SYM, where non-abelian exponentiation forbids the presence of $1\/\\epsilon^2$ in vertex or bubble graphs. In order to proceed and extract a generalized potential, taking properly into account the one-loop and two-loop results, we need an exponentiation ansatz: we propose the following form for the unrenormalized loop\n\\begin{equation}\n\\mathcal{W}_{+}=\\frac{M \\exp(V_{N})+ N \\exp(V_{M})}{N+M}. \n\\end{equation}\nIt is not difficult to check that our results are compatible with this  double-exponentiation where \n\\begin{align}\n\\label{unrenpot}\nV_{N}=&\\left(\\frac{2\\pi}{\\kappa}\\right) N\\left(\\frac{ \\Gamma(1\/2-\\epsilon)}{4\\pi^{3\/2-\\epsilon}}\\right)\n(\\mu L)^{2\\epsilon}\\left[\\frac{1}{\\epsilon}\\left(\\frac{\\cos\\frac{\\theta}{2}}{\\cos\\frac{\\varphi}{2}}-2\\right)-2\\frac{\\cos\\frac{\\theta}{2}}{\\cos\\frac{\\varphi}{2}}\\log \\left(\\sec \\left(\\frac{\\varphi }{2}\\right)+1\\right)\\right]+\\nonumber\\\\\n&+N^2\\left(\\frac{2\\pi}{\\kappa}\\right)^{\\!\\!2}\\!\\frac{\\Gamma^{2}\\!\\left(\\frac{1}{2}-\\epsilon\\right)}{16 \\pi^{3-2\\epsilon}}(\\mu L)^{4\\epsilon}\\left[\\frac{1}{\\epsilon} \\log\\left(\\cos^{2}\\frac{\\varphi}{2}\\right)\\left(\\frac{\\cos\\frac{\\theta}{2}}{\\cos\\frac{\\varphi}{2}}-1\\right)+O(1) \\right]\n\\end{align}\nThe generalized potential $V_{M}$ is of course obtained by exchanging $M$ with $N$ in the above formula. We remark that the actual exponentiation of the one-loop term is a non-trivial support of our assumption and of the correctness of our computations, involving a delicate balance between exchanging and interacting contributions. From the physical point of view we could also justify the presence of two generalized potentials, simply recalling that we have two different test particles running in our contour. Following \\cite{Lee:2010hk} it is straightforward to show that in $U(N)\\times U(M)$ ${\\cal N}=6$ theories two kinds of particles arise from the relevant higgsing procedure and which transform respectively in the $({\\bf N},{\\bf 1})$ and $({\\bf 1},{\\bf M})$ representations and their conjugate, that we call $W_N$ and $W_M$ bosons. It is clear that a  pair of  $W_N$ and $W_M$ cannot form a singlet of the color indices and there is no generalization of the quark-antiquark potential in this case. On the other hand a pair of $W_N\\bar{W}_N$ or $W_M\\bar{W}_M$ do form color singlets, hence there are two potentials in this theory.\n\\subsection{The renormalized generalized potentials}\n\\label{outcome}\n The outcome  of our extensive two-loop computation  contains some  puzzling unexpected features which deserve a more detailed analysis. To begin with,  let us consider  the one-loop contribution in \\eqref{unrenpot}.\nWhen $\\theta=\\varphi=0$ our cusp degenerates into a segment of length $2L$  with the couplings of the $1\/2$ BPS straight line and its ({\\it unrenormalized}) value is given at the first non-trivial order by \n \\begin{align}\n \\label{Ren1}\n{W}^{(1)}_{\\rm line}=&-\\left(\\frac{2\\pi}{\\kappa}\\right)\\frac{M N}{N+M}\\left(\\frac{ \\Gamma(1\/2-\\epsilon)}{4\\pi^{3\/2-\\epsilon}}\\right)\n(\\mu L)^{2\\epsilon}\\left[\\frac{1}{\\epsilon}+2\\log \\left(2\\right)+O(\\epsilon)\\right]=\\nonumber \\\\\n=&-\\left(\\frac{2\\pi}{\\kappa}\\right)\\frac{M N}{N+M}\\left(\\frac{ \\Gamma(1\/2-\\epsilon)}{4\\pi^{3\/2-\\epsilon}}\\right)\n\\frac{(2 L\\mu)^{2\\epsilon}}{\\epsilon}+O(\\epsilon).\n\\end{align}\nThis divergent result appears to contradict the  expectation  that the  $1\/2-$BPS straight-line  is trivial ({\\it i.e.} equal to $1$) as  occurs in $\\mathcal{N}=4$ SYM.  In that case an analogous computation  for a segment \nof length $2L$ in Feynman gauge  would have led to an exact cancellation between the gauge and the scalar contribution yielding as final result  $W^{(1)}_{\\rm line}=0$. We remark, however, that this manifest zero in $\\mathcal{N}=4$ SYM is peculiar  of the Feynman gauge. In a generic \n$\\alpha-$gauge  the cancellation is only partial and a divergent term similar to  \\eqref{Ren1} survives,\n\\begin{equation}\n\\label{N=4div}\nW^{(1)}_{\\rm line}=g^2N(1-\\alpha) \\frac{\\Gamma(1-\\epsilon)}{16\\pi^{2-\\epsilon}}\\frac{(2  L\\mu)^\\epsilon}{\\epsilon}. \n\\end{equation}\nThe $\\alpha-$dependence in \\eqref{N=4div} signals  that we are dealing with a gauge-dependent divergence\\footnote{The gauge origin of these additional divergences  is even more transparent when  we consider a circular sector of aperture $2\\pi -\\theta$ in $\\mathcal{N}=4$.  There, next to  the expected result in Feynman gauge, there is  a divergent term given by  \\[g^2N(1-\\alpha) \\frac{\\Gamma(1-\\epsilon)}{16\\pi^{2-\\epsilon}} \\frac{(4\\sin^{2}\\frac{\\theta}{2})^\\epsilon}{\\epsilon}.\\]\nWhen we close the circle ($\\theta=0$),  thus recovering the gauge invariant operator, the coefficient of the  divergence simply vanishes.}, but this is not surprising. In fact the result \\eqref{N=4div}  is the expectation  value for a segment of length $2L$, which  does not define a gauge invariant operator unless $L=\\infty$.  However the limit $L\\to \\infty$ is delicate and it cannot be taken before  renormalizing the finite length operator.\n\nThe systematic renormalization of Wilson operator on open contours is a subject widely discussed  in the literature\n\\cite{Dorn1,Dorn2,Aoyama:1981ev,Knauss:1984rx,Dorn:1986dt}\nand an exhaustive presentation of the topic is beyond the goal  of this paper. Below we shall recall some general facts  using  YM  or $\\mathcal{N}=4$ SYM  as our  pedagogical examples.  The case of ABJM will be considered later.\n\nAn efficient frame-work  for discussing the renormalization of path ordered phase factors was introduced \nby \\cite{Arefeva:1980zd,Gervais:1979fv}. In this approach these non-local operators  are represented as \nthe  two point function\n$\\left\\langle\\psi(-L)\\bar\\psi(L)\\right\\rangle_{0}$ of the  one-dimensional fermionic  {\\it bare} action  \\footnote{For open loops the action must also contains boundary terms (see  {\\it e.g.} \\cite{Dorn:1986dt}) but for simplicity we shall neglect them.}\n\n\\begin{equation}\n\\label{Spsi}\nS=\\int_{-L}^{L}\\!\\! dt ~\\bar\\psi(i\\partial_{t}+  g \\mathcal{A}_{\\mu}\\dot{x}^{\\mu} )\\psi\n\\end{equation}\nwhere $\\mathcal{A}_{\\mu}$ stands for the connection of which we are computing the quantum holonomy.  \nThe familiar techniques of renormalization for local Green function can be therefore applied. \n\nIn  this language  the  divergence \\eqref{N=4div}  is responsible  for the familiar  wave-function renormalization of the field $\\psi$ and it can be in fact eliminated by introducing $\\psi_{R}= Z^{-1\/2}_{\\psi}\\psi$.  This interpretation is also consistent with the fact  that its value is gauge-dependent. The usual \nperimeter divergence, present in a cut-off regularization,  appears as a mass counter-term for the spinor $\\psi$\n in the renormalized action. \n\nAccording to the previous discussion, the renormalized  operator  for an open smooth contour $C$ is obtained as \\cite{Dorn1,Dorn2,Aoyama:1981ev,Knauss:1984rx,Dorn:1986dt}\n\\begin{equation}\n\\label{ren-open}\n\\begin{split}\n{\\cal W}_{\\rm ren.}=&\\left\\langle\\psi_{R}(-L)\\bar\\psi_{R}(L)\\right\\rangle_{0}=\\\\\n=&Z^{-1}_{\\psi}\n\\left\\langle\\psi(-L)\\bar\\psi(L)\\right\\rangle_{0}=Z^{-1}_{\\psi} e^{-\\ell \\delta m}\\left\\langle\nP\\exp\\left(i g_{\\rm ren.} \\frac{Z_{1}}{ Z_{3}}\\int_{C} dx^{\\mu} \\mathcal{A}^{\\rm ren.}_{\\mu}\\right)\\right\\rangle_{0},\n\\end{split}\n\\end{equation}\nwhere $Z_{1}$ and $Z_{3}$ are the usual renormalization for the gauge coupling constant and the wave-function renormalization for $\\mathcal{A}_{\\mu}$. Moreover $\\ell$ is the perimeter of the smooth open contour $C$; the mass renormalization $\\delta m$ is zero when dimensional regularization is used since it corresponds to a power-like divergence.\n\nAn important remark is now in order.  In dimensional regularization the new renormalization constant $Z_{\\psi}$ can be shown to be  independent of the shape of the  smooth contour \\cite{Dorn1,Dorn2,Aoyama:1981ev,Knauss:1984rx,Dorn:1986dt}  (up to  a redefinition of the renormalization scale). Accordingly its value can be computed for a  finite segment and then used for other smooth  contours.\n\nWhen we close the circuit, thus considering a Wilson loop,   a new divergence appears  \\cite{Dorn1,Dorn2,Aoyama:1981ev,Knauss:1984rx,Dorn:1986dt}, since the two fields  in $\\left\\langle\\psi(-L)\\bar\\psi(L)\\right\\rangle_{0}$ are now located at the same point.  More correctly  the closed loop does not define a two-point function, but the expectation value of a composite operator: this explains the need of a further renormalization.   However the effect of this additional ingredient  is to exactly cancel  the factor \n$Z^{-1}_{\\psi}$ \\cite{Dorn1,Dorn2,Aoyama:1981ev,Knauss:1984rx,Dorn:1986dt} and one recovers the \nfamiliar  and simple result\\footnote{This result was first shown in \\cite{Exp}  using combinatorial techniques.}\n\\begin{equation}\n\\begin{split}\n\\label{ren-closed}\n{\\cal W}^{\\rm clos.~ loop}_{\\rm ren.}= e^{-\\ell \\delta m}\\left\\langle\nP\\exp\\left(i g_{\\rm ren.} \\frac{Z_{1}}{ Z_{3}}\\oint_{C} dx^{\\mu} \\mathcal{A}^{\\rm ren.}_{\\mu}\\right)\\right\\rangle_{0},\n\\end{split}\n\\end{equation}\n{\\it i.e.}  a smooth Wilson loop does not contain any new divergence with respect to  those of the gauge theory, apart from the one proportional to the perimeter of the contour.  \nSince $Z_{\\psi}$  is only present when dealing with open circuits, but disappears for closed loops,  it  is also named  $Z_{\\rm open}$. \n\nLet us remark that  the final equalities in \\eqref{ren-open} and \\eqref{ren-closed}  define a procedure for  renormalizing  smooth  path ordered phase factors independently of the fermionic representation used to prove them.\n\nWe come back  to the example of the segment in $\\mathcal{N}=4$ SYM. If we introduce the wave-function\nrenormalization \n\\begin{equation}\nZ_{\\rm open}=1+g^2N(1-\\alpha) \\frac{\\Gamma(1-\\epsilon)}{16\\pi^{2-\\epsilon}} \\frac{(2  L\\mu)^\\epsilon}{\\epsilon}. \n\\end{equation}\nthe expectation value of the renormalized operator becomes again trivial as occurs in Feynman gauge. \nIn the case of ABJM, the divergence  can be handled in the same way  by introducing\n \\begin{align}\nZ_{\\rm open}\n=&1-\\left(\\frac{2\\pi}{\\kappa}\\right)\\frac{M N}{N+M}\\left(\\frac{ \\Gamma(1\/2-\\epsilon)}{4\\pi^{3\/2-\\epsilon}}\\right)\n\\frac{(2 L\\mu)^{2\\epsilon}}{\\epsilon}+O(\\epsilon).\n\\end{align}\nIn other words, with respect to the familiar $\\mathcal{N}=4$ result ($\\alpha=1$), the Landau gauge used to compute \\eqref{Ren1} in ABJM theories   does not  enjoy  the simplifying property $Z_{\\rm open}=1$.\n\n\\noindent\n Let us now turn to piecewise smooth contours \\cite{Polyakov:1980ca,Korchemsky:1987wg,Dorn1,Dorn2,Knauss:1984rx,Dorn:1986dt}, namely to contours containing points where the derivative $\\dot{x}^{\\mu}$ is  discontinuous. If there is  a {\\it cusp} at  $t=t_{0}$, {\\it i.e.} $\\lim_{t\\to t_{0}^{+}} \\dot{x}^{\\mu}(t)\\ne \\lim_{t\\to t_{0}^{-}} \\dot{x}^{\\mu}(t)$, the renormalization of the action \\eqref{Spsi} requires \n an additional counter-term proportional to  $\\bar \\psi(t_{0}) \\psi(t_{0})$ \\cite{Knauss:1984rx,Dorn:1986dt}.   To argue the origin of this new counter-term we observe that a reasonable renormalization procedure  should respect the composition rule for path-ordered  phase factors on smooth contours. Specifically if we split a regular contour $C$ $\\{ x(t) | -L\\le t\\le L\\}$ into two  sub-contours $C_{1}$ $\\{ x(t) | -L\\le t\\le t_{0}\\}$  and $C_{2}$ $\\{ x(t) | t_{0}\\le t\\le L\\}$ \n\\begin{equation}\n\\mathcal{W}^{\\rm ren.} (C_{1})\\mathcal{W}^{\\rm ren.}(C_{2})= W^{\\rm ren.}(C).\n\\end{equation}\n In terms of the two point function of the one-dimensional  fermion $\\psi$  this property reads\n \\begin{equation}\n \\langle\\psi(-L)\\bar\\psi(t_{0})\\rangle\\langle\\psi(t_{0})\\bar\\psi(L)\\rangle=\n  \\langle\\psi(-L)(\\bar\\psi\\psi)(t_{0})\\bar\\psi(L)\\rangle=\\langle\\psi(-L)\\bar\\psi(L)\\rangle.\n \\end{equation}\n The intermediate equality implies that  the renormalization factor $Z_{\\bar\\psi\\psi}$ for the composite operator $(\\bar\\psi\\psi)(t_{0})$ is $1$. This is an equivalent manifestation  of the previous statement that $Z_{\\rm open}$  drops out  when the two endpoints of a loop are joined  smoothly.  If $t_{0}$ is instead the position of a cusp the factor $Z_{\\bar\\psi\\psi}$  can be in general different from $1$ and it must be included  in the renormalization of the Wilson-operator. Its insertion leads to the following modification of \\eqref{ren-open} for open contour with one cusp\n \\cite{Dorn1,Dorn2,Knauss:1984rx,Dorn:1986dt}\n \\begin{equation}\n \\label{Wcusp}\n\\begin{split}\n{\\cal W}_{\\rm ren.}=Z^{-1}_{\\rm open} Z_{\\bar\\psi\\psi} e^{-\\ell \\delta m}\\left\\langle\nP\\exp\\left(i g_{\\rm ren.} \\frac{Z_{1}}{ Z_{3}}\\int_{C} dx^{\\mu} \\mathcal{A}^{\\rm ren.}_{\\mu}\\right)\\right\\rangle_{0}.\n\\end{split}\n\\end{equation}\nIn the following we shall replace the symbol $Z_{\\bar\\psi\\psi}$ with the more familiar $Z_{\\rm cusp}$.\n\nThe renormalization factor $Z_{\\rm cusp}$ can be shown to depend only on the angle $\\varphi$ of the cusp  and not on the global geometry of the circuit, and to be gauge invariant.  Moreover it must satisfy a simple renormalization condition \\cite{Dorn2,Knauss:1984rx,Dorn:1986dt}\n\\begin{equation}\n\\label{rescind}\n\\left. Z_{\\rm cusp}\\right|_{\\varphi=0}=1,\n\\end{equation}\nsince  the cusp disappears for $\\varphi=0$ and no new renormalization is needed  apart from $Z_{\\rm open}$.  This condition also appears in \\cite{Korchemsky:1987wg}  as a   Ward identity  for the vertex  in the one-dimensional field theory. The new  factor will give  origin to the  well-known cusp-anomalous dimension, which is defined through the relation \n\\begin{equation}\n\\label{r1bps}\n\\gamma=\\mu\\frac{d}{d\\mu} \\log Z_{\\rm cusp}.\n\\end{equation}\nWe expect that the above renormalization procedure   carries over to the case of the Wilson loop in  ABJ theory  with minor changes.  In fact  the structure of eq. \\eqref{Wcusp}  is substantially independent  of the specific form of $\\mathcal{A}_{\\mu}$ and  of the route used to prove it.  A detailed\nproof of the above results, in the case of the phase operator defined by the super-connection \\eqref{superconnection}, could be obtained  by using   the supersymmetric quantum \nmechanics  discussed in \\cite{Lee:2010hk} as a starting point instead of  \\eqref{Spsi}.   An obvious difference with the above discussion arises when considering the  renormalization \ncondition. For our operators  eq. \\eqref{rescind} must be  replaced by\n\\begin{equation}\n\\label{r1abps}\n\\left. Z_{\\rm cusp}\\right|_{\\varphi=\\theta=0}=1.\n\\end{equation}\nRecall, in fact, that we have also a cusp in the $R-$symmetry directions  governed by the angle $\\theta$ next to geometrical one  given by $\\varphi$.  In this language  the BPS condition $\\theta=\\varphi$  should translate into the following \n\\begin{equation}\n\\label{r2bps}\n\\left. Z_{\\rm cusp}\\right|_{\\varphi=\\theta}=1.\n\\end{equation} \nEq. \\eqref{r2bps} is not equivalent to \\eqref{r1abps}. Thus  the BPS condition still provides a check of the correctness of our computation.\n\n\\noindent\nHaving in mind  the above discussion, it is straightforward  to extract the renormalized generalized potential $V^{\\rm Ren.}_{N}$ from \\eqref{unrenpot}. We obtain\n\\begin{align}  \n\\!\\!V^{\\rm Ren.}_{N}=&\\left(\\frac{2\\pi}{\\kappa}\\right) N\\left(\\frac{ \\Gamma(\\frac{1}{2}-\\epsilon)}{4\\pi^{3\/2-\\epsilon}}\\right)\\!\n(\\mu L)^{2\\epsilon}\n\\!\\!\\left[\\frac{1}{\\epsilon}\\left(\\frac{\\cos\\frac{\\theta}{2}}{\\cos\\frac{\\varphi}{2}}-1\\right)\\!\\!-2\\frac{\\cos\\frac{\\theta}{2}}{\\cos\\frac{\\varphi}{2}}\\log \\left(\\sec \\left(\\frac{\\varphi }{2}\\right)\\!+\\!1\\right)\\!\\!+\\log 4\\right]\\!+\\nonumber\\\\\n&+\\left(\\frac{2\\pi}{\\kappa}\\right)^{\\!\\!2}N^2\\!\\left(\\frac{\\Gamma\\!\\left(\\frac{1}{2}-\\epsilon\\right)}{4 \\pi^{3\/2-\\epsilon}}\\right)^2(\\mu L)^{4\\epsilon}\\left[\\frac{1}{\\epsilon}\\log\\left(\\cos\\frac{\\varphi}{2}\\right)^2\\left(\\frac{\\cos\\frac{\\theta}{2}}{\\cos\\frac{\\varphi}{2}}-1\\right)+O(1)\\right],\n\\end{align}\nwhere we have included the finite terms for completeness. The terms proportional to $1\/\\epsilon$ give the logarithm of the celebrated  $Z_{\\rm cusp}$.  It is trivial to check that $\\left. Z_{\\rm cusp}\\right|_{\\varphi=\\theta}=1$.\n\n \nThe quark-antiquark potential is recovered by taking the limit $\\varphi\\to \\pi$ and following the prescription of \\cite{Drukker:2011za}\n\\begin{equation}\nV^{(s)}_{N}(R)=\\frac{N}{k}\\frac{1}{R}-\\left(\\frac{N}{k}\\right)^2\\frac{1}{R}\\log\\left(\\frac{T}{R}\\right).\n\\end{equation}\n\nWe observe a logarithmic, non-analytic term in $T\/R$ at the second non-trivial order that, as in four dimensions, is expected to disappear when resummation of the perturbative series is performed. We can also perform the opposite limit, taking large imaginary $\\varphi$, and we recover the universal cusp anomaly \n\\begin{equation}\n\\gamma_{cusp}=\\frac{N^2}{k^2},\n\\end{equation}\nthat is the result obtained directly from the light-like cusp \\cite{Henn:2010ps}.\n\\subsection{$1\/2-$BPS line versus $1\/6-$BPS line}\n\\label{versus}\nIn the previous subsection we have discussed the appearence of spurious divergences in the quantum computation of our cusped Wilson loops and explained their subtraction procedure: we have also remarked that these divergences obstinately persits in the case of $1\/2-$BPS straight-lines, although not contradicting their triviality. However there is still an additional feature \nthat may appear puzzling. In \\cite{Drukker:2009hy} it was pointed out that the $1\/2-$BPS straight-line is cohomologically equivalent to its $1\/6-$BPS counterpart, defined in \\cite{Drukker:2008zx}.  One can easily show that, at least at one loop, the  expectation value of the latter is trivial without requiring any renormalization, exactly as in ${\\cal N}=4$: encountering  divergences in  the evaluation of $1\/2-$BPS straight-line seems therefore to contradict the cohomological equivalence. \n\nThe key point of \\cite{Drukker:2009hy}, in order  to establish the equivalence of the two observables, was to observe that the difference between \n$\\mathcal{W}^{1\/2}_{\\rm line}$ and $\\mathcal{W}^{1\/6}_{\\rm line}$ can be cast into a $Q-$exact term\n\\begin{equation}\n\\label{pip}\n\\mathcal{W}^{1\/2}_{\\rm line}-\\mathcal{W}^{1\/6}_{\\rm line}=Q V,\n\\end{equation} \nwhere  the supercharge  $Q$ is that  generated by the spinor  $ \\bar\\theta^{IJ\\beta}= (\\bar n^{I} \\bar w^{J}-\\bar n^{J} \\bar w^{I})\\bar\\eta^{\\beta} -i\\epsilon^{IJKL} n_{K} w_{L}\\eta^{\\beta}$, while  the scalar couplings in  $\\mathcal{W}^{1\/6}_{\\rm line}$ are governed  by  the matrices $M_{J}^{\\ \\ I}=\n\\widehat M_{J}^{\\ \\ I}= \\delta^{I}_{J}-2  n_{J} \\bar n^{I}-2  w_{J} \\bar w^{I}.$  A complete expression for $V$ has been presented in  \\cite{Drukker:2009hy}, but we shall not report it  here.   To understand why the above identity fails, it will suffice to consider its lowest non trivial order  in $1\/k$: using the notation of \\cite{Drukker:2009hy} we explicitly obtain\n\\begin{equation}\n\\label{pip1}\n{\\small\n\\begin{split}\n-i\\int_{-\\infty}^{\\infty} \\!\\!\\!\\! \\!\\!d t~ \\tilde{\\mathcal{L}}_{B}-\\int_{-\\infty}^{\\infty} \\!\\!\\!\\! \\!\\!dt_{1}\\int_{-\\infty}^{t_{1}} \\!\\!\\!\\! \\!\\!d t_{2}\n\\mathcal{L}_{F}(t_{1})\\mathcal{L}_{F}(t_{2})=-\\frac{1}{2} Q\\left(\\int_{-\\infty}^{\\infty}\\!\\!\\!\\! \\!\\! dt_{1}\\int_{-\\infty}^{t_{1}}\\!\\!\\!\\! \\!\\! d t_{2}[ \\Lambda(\\tau_{1}) \\mathcal{L}_{F}(\\tau_{2})-\\mathcal{L}_{F}(\\tau_{1}) \\Lambda(\\tau_{2})) ]\\!\n\\right)\\!\\!.\n\\end{split}}\n\\end{equation}\nThe quantities $\\tilde{\\mathcal{L}}_{B},~\\mathcal{L}_{F}$ and $\\Lambda $ in \\eqref{pip1}  are defined by the following matrices\n\\begin{equation}\n{\\small\n\\begin{split}\n\\!\\!\\!\\tilde{\\mathcal{L}}_{B}=- \\frac{4 \\pi i }{k}  |\\dot x| \\begin{pmatrix}\nC_{\\bar w}\\bar C^{w}\n&0\\\\\n0 &\n\\bar C^{w}C_{\\bar w}\n\\end{pmatrix} \n\\  \\ \n\\mathcal{L}_{F}=-i \\sqrt{\\frac{2\\pi}{k}}  |\\dot x| \\begin{pmatrix}\n0\n& \\eta\\bar\\psi\\\\\n   \\psi\\bar{\\eta} &\n0\n\\end{pmatrix} \n\\ \\ \n \\Lambda=-\\frac{1}{2} \\sqrt{\\frac{2\\pi}{k}}|\\dot x|\\begin{pmatrix} 0 &  i\\bar C^{w}\\\\   C_{\\bar w} & 0 \\end{pmatrix}\n \\end{split}}\n\\end{equation}\nwhere  the scalars are given by $C_{\\bar w}= \\bar w^{I} C_{I}$ and $\\bar C^{w}= \\bar C^{J} w_{J}$, the reduced  spinors  are written as  $\\bar\\psi=\\bar\\psi^{I} n_{I}$  and $\\psi=\\psi_{I}\\bar n^{I}$.\n\nWhen we replace the  infinite straight-line with a segment of length  $2L$ to tame the infrared divergences,\nthe above equality receives a correction from the  value of the scalar fields on the  boundary. Taking properly into account some total derivatives, usually discarded for infinite lenght, \\eqref{pip1} is replaced by\n\\begin{equation}\n\\label{pip3}\n{\\small\n\\begin{split}\n\\int_{-\\infty}^{\\infty} \\!\\!\\!\\! \\!\\!d t~ &\\tilde{\\mathcal{L}}_{B}+\\int_{-\\infty}^{\\infty} \\!\\!\\!\\! \\!\\!dt_{1}\\int_{-\\infty}^{t_{1}} \\!\\!\\!\\! \\!\\!d t_{2}\n\\mathcal{L}_{F}(t_{1})\\mathcal{L}_{F}(t_{2})+4 i \\int_{-L}^{L}\\!\\!\\!\\!d t \\left(\\frac{\\Lambda(L) \\Lambda(t)}{|\\dot{x}_{L}|}+\\frac{\\Lambda(t) \\Lambda(-L)}{|\\dot{x}_{-L}|}\\right)=\\\\\n&=-\\frac{1}{2} Q\\left(\\int_{-L}^{L}\\!\\!\\!\\! \\!\\! dt_{1}\\int_{-L}^{t_{1}}\\!\\!\\!\\! \\!\\! d t_{2}[ \\Lambda(\\tau_{1}) \\mathcal{L}_{F}(\\tau_{2})-\\mathcal{L}_{F}(\\tau_{1}) \\Lambda(\\tau_{2})) ]\\!\n\\right).\n\\end{split}}\n\\end{equation}\nIn other words, if defined on a segment the two Wilson operator are not cohomologically equivalent! Actually we can go further and observe that the divergence of the $1\/2-$BPS line comes entirely from these {\\it boundary terms}, when evaluated at quantum level. For instance it is easy to check that the new term \nin \\eqref{pip3} is accountable for  the  result \\eqref{Ren1}. The renormalization procedure described in the \nprevious subsection is built to subtract exactly these spurious contributions.\n\n\n\n\n\n  \n  \n  \n  \n  \n  \n\\section{Conclusions and outlook}\nIn this paper we have studied a family of cusped Wilson loops in ABJ(M) super Chern-Simons theory, constructed from two 1\/2 BPS lines implying the presence of peculiar fermionic  couplings \\cite{Drukker:2009hy}. They depend on two parameters, $\\varphi$ and $\\theta$, that describe the geometrical and $R$-symmetry angles, respectively, between the two rays. We have studied the supersymmetric properties of these configurations and their relation with closed contours, obtained through conformal transformations. Different limits on the parameters allow to reach interesting observables, as the analogous of the quark-antiquark potential or the universal cusp anomalous dimension. We have performed an explicit two-loop computation in dimensional regularization and we have obtained the divergent part of these contour operators. Our results suggest the existence of two generalized potentials in this theory and, after renormalization, we have obtained in the relevant limits the universal cusp anomaly and the $W_{N(M)}\\bar{W}_{N(M)}$ binding energy. \n\nThe construction of a generalized potential from a cusped Wilson loop opens many interesting possibilities in ABJ(M) theory: one could try to compute the radiation of a particle moving along an arbitrary smooth path, as done in ${\\cal N}=4$ SYM \\cite{Correa:2012at}. Further, one could hope to find a three dimensional analogue of the set of TBA integral equations, recently discovered in \\cite{Drukker:2012de,Correa:2012hh}, describing non-perturbatively the D=4 generalized cusp (see also \\cite{Bykov:2012sc,Henn:2012qz,Gromov:2012eu} for very recent developments). It is also tempting to speculate on the possibility to derive the infamous interpolating function $h(\\lambda)$ \\cite{H,Grignani:2008is,Nishioka:2008gz,BT1,BT2,MZ,GGY}, by comparing the integrability computations with exact results obtained through localization \\cite{Kapustin:2009kz}. An important step in this program would be the derivation of general class of Wilson loops with lower degree of supersymmetry, specifically some analogue of the DGRT loops \\cite{Drukker:2007qr} in ${\\cal N}=6$ super Chern-Simons theory. A particular case, the wedge on $S^2$, has been discussed here in sec. 3: a general construction of BPS loops on $S^2$, preserving fractions of supersymmetry, will be presented soon \\cite{CGMS}. It would be of course important to compute their quantum expectation value at weak coupling, by perturbation theory, and at strong coupling, using string techniques. Hopefully their exact expression could be derived through localization methods.    \n\n\n\n\\section*{Acknowledgements}\nThis work was supported in part by the MIUR-PRIN contract 2009-KHZKRX. We warmly thank Antonio Bassetto, Valentina Cardinali, Valentina Forini, Valentina Giangreco Marot\\-ta Puletti and especially Nadav Drukker for useful discussions.\n\n\\newpage\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
{"text":"\\section{Introduction: The Metrization\nProcedure}\n\nBarbilian's metrization procedure was introduced in \\cite{B1934}\nand it was the subject of an inspiring correspondence between D.\nBarbilian and W. Blaschke \\cite{B1940} in 1934 and thereafter. The\ntheory received  a larger audience due to P. Kelly \\cite{K1954}\nand a major development due to D. Barbilian\n\\cite{B1959a,B1959b,B1960,BR1962}. Over the years, the paper\n\\cite{B1934} has been cited many times. Recent studies are due to\nA.F. Beardon \\cite{Beardon1998},  F. Gehring and K. Hag\n\\cite{GH2000}, as well as P. H\\\"ast\\\"o, Z. Ibragimov and other\nauthors \\cite{H2003, H2003b, H2004b, H2004, H2005b,\nH2005,HL2004,HPS2005, I2002, I2003a, I2003b}.\n The geometric viewpoint is discussed in the\nmonograph \\cite{WB1996carte}. All of these works cite and have a\ncommon source in Barbilian's paper \\cite{B1934}. The examples\nexplored in the present work aim to discuss Barbilian's\nmetrization procedure in the context of its relations with various\nclasses of metrics, as for example Riemann, Finsler, Lagrange or\nLagrange generalized metrics (see \\cite{BCS2000, MAB}).\n\nThe following construction is given by Barbilian \\cite{B1959a} and\nit is the development of the idea from \\cite{B1934}. Consider two\narbitrary sets $K$ and $J.$ The function $f : K \\times J\n\\rightarrow \\mathbb{R}_+^*$ is called an influence  of the set $K$\nover $J$ if for any $A,B \\in J$ the ratio\n$g_{AB}(P)=\\frac{f(P,A)}{f(P,B)}$ has a maximum $M_{AB} \\in\n\\mathbb{R}$ when $P \\in K.$ Note that $g_{AB}:K \\rightarrow\n\\mathbb{R}_+^*.$ In \\cite{B1959a} it is pointed out that if we\nassume the existence of $\\max g_{AB}(P),$ when $P \\in K,$ then\nthere also exists $m_{AB} = \\min_{P \\in K} g_{AB}(P) = \\frac{1}{\nM_{BA}}.$\n\nFor example (see \\cite{B1959a}), if $T$ is a topological space,\n$K$ a compact subset in $T$, and $J$ some arbitrary subset,then\nany function $f: K \\times J \\rightarrow \\mathbb{R}_+^*$ continuous\nin the first argument is an influence on $J.$ It is known since\n\\cite{B1959a} that $d: J \\times J \\rightarrow \\mathbb{R}_+$ given\nby\\begin{equation}\\label{metrica1}d(A,B) = \\ln \\frac{\\max_{P\\in K}\ng_{AB}(P)}{\\min_{P\\in K} g_{AB}(P)}\\end{equation}is a\nsemidistance, i.e.: (1) if $A=B$ then $d(A,B)=0;$ (2) $d$ is\nsymmetric; (3) $d$ satisfies triangle inequality.\n\nThe influence $f: K \\times J \\rightarrow \\mathbb{R}_+^*$ is called\neffective if there is no pair $(A,B) \\in J \\times J$ such that the\nratio $g_{AB}(P)=\\frac{f(P,A)}{f(P,B)}$ is constant for all $P \\in\nK.$ In \\cite{B1959a} it is shown that if $f: K \\times J\n\\rightarrow \\mathbb{R}_+^*$ is an effective influence, then\n(\\ref{metrica1}) is a distance.\n\n\\section{Examples}\n\\begin{example} Barbilian's\nmetrization procedure yields the  Euclidean distance in a plane $(\n\\pi )$ in $\\mathbb{R}^3,$ if we consider a plane $(\\delta)$\nparallel to the plane $(\\pi)$ and take $J=(\\pi), K=(\\delta)$, and\nthe influence function\n $f: K \\times J \\rightarrow \\mathbb{R}_+^*$, $f(M,A)=\n\\exp \\circ \\left[ \\frac{1}{2} || (Pr \\times Id) (M,A)|| \\right] =\ne^{\\frac{1}{2}||M'A||}.$\\end{example}\n\n\\begin{example}\nBarbilian's metrization procedure yields the spherical distance in\na complete sphere in $\\mathbb{R}^3.$ \\end{example}\n\nTo see this, consider two concentric spheres $S_1$ and $S_2$ in\n$\\mathbb{R}^3,$ and let their common center be $O.$ We take\n$S_1=K$ and $S_2=J,$ and  $A,B \\in J$ and $M\\in K.$ Denote by\n$\\{M'\\}=(OM \\cap J$ and define $Pr$ the radial projection from\n$S_1$ to $S_2$ given by $Pr(M)=M'.$ Denote by $( \\ . \\ )$ the\nspherical distance, and consider the influence function $f: K\n\\times J \\rightarrow \\mathbb{R}_+^*$, $f(M,A)=\\exp \\circ\n\\left[\\frac{1}{2} ( (Pr \\times Id) (M,A) ) \\right] =\ne^{\\frac{1}{2}(M'A)}.$\n\nThus, Barbilian's metrization procedure can generate Riemannian\nmetrics. Our goal is to show that Barbilian's metrization\nprocedure generates, for other choices of $K,J,$ and $f$, Lagrange\ngeneralized metrics not reducible to a Riemannian, Finslerian or\nLagrangian metric.\n\nTo complete our discussion, we mention here the following result,\nneeded in the remaining part of this section. This is a particular\nform of the result from \\cite{B1960}, part 2, paragraph 7, and a\nversion of the argument used in \\cite{H2004} in the proof of Lemma\n3.5.\n\\begin{lemma}\\label{mimetic} Let $K$ and $J$ be two\nsubsets of the Euclidean plane $\\mathbb{R}^2,$ and $K = \\partial\nJ.$ Consider the influence $f(M,A)= ||MA||,$ where by $||MA||$ we\ndenote the Euclidean distance. Consider $$g_{AB} (M) =\n\\frac{f(M,A)}{f(M,B)} = \\frac{||MA||}{||MB||}$$ and consider the\ndistance induced on $J$ by the Barbilian's metrization procedure,\n$d^B(A,B).$ Suppose furthermore that for $M \\in K$ the extrema\n$\\max g_{AB}(M)$ and $\\min g_{AB}(M)$ for any $A$ and $B$ in $J$\nare attained each in an unique point in $K.$ Then:\n\n(a) For any $A \\in J$ and any line $d$ passing through $A$ there\nexist exactly two circles tangent to $K$ and to $d$ in $A.$\n\n(b) The metric induced by the Barbilian distance has the form\n\\begin{equation}\\label{ecuatia_mimetica}ds^2 = \\frac{1}{4}\n\\left(\\frac{1}{R}+\\frac{1}{r} \\right)^2 (dx_1^2 +\ndx_2^2),\\end{equation} where $R$ and $r$ are the radii of the\ncircles described in (a).\\end{lemma}\n\n\\begin{example} Let $K=$ be the line $\\{\ny=0 \\}$ in the $xy$-plane. Let $J= \\{ (x,y) \/ y>0 \\}.$ Take the\nfunction $||MA||$ as influence. Then the associated ratio is\n$f(M)=\\frac{||MA||}{||MB||}.$ By applying Barbilian's metrization\nprocedure, we only need to analyze the existence of minimum and\nmaximum for the function$$g(x) = \\frac{x^2 - 2x_0 \\cdot x + x_0^2\n+ y_0^2}{x^2 - 2 x_1 \\cdot x + x_1^2 +y_1^2}.$$ A straightforward\napplication of Lemma \\ref{mimetic} yields, after computations $$R\n= \\frac{y \\sqrt{m^2+1}}{-1+\\sqrt{m^2+1}}$$ and $$r =\n\\frac{y\\sqrt{m^2+1}}{1+\\sqrt{m^2+1}},$$ that is $\\frac{1}{4}\n\\left(\\frac{1}{R}+\\frac{1}{r} \\right)^2=\\frac{1}{y^2},$ i.e.\n$ds^2=\\frac{1}{y^2} (dx^2+dy^2),$ which is the Poincar\\'e metric\non the upper half-plane.\\end{example}\n\n\\begin{example}Consider\n$\\mathbb{R}^2$ endowed with the Euclidean distance $||.||$. It is\nknown from \\cite{B1959a} that for any circle $K$ of radius $\\rho$\nin $\\mathbb{R}^2$, and for $J$ the interior of $K$, a Barbilian's\ndistance is obtained in $J$ by taking the influence $f(P,A)=\n||PA||.$ For a given point $(x,y)$ in J and for an arbitrary line\nof slope $m$ passing through $(x,y),$ we find $$R =\n\\frac{\\sqrt{m^2+1}}{2} \\cdot \\frac{\\rho^2 - x^2 -\ny^2}{\\rho\\sqrt{m^2+1} - x m +y}.$$ Similarly, we get, $$r =\n\\frac{\\sqrt{m^2+1}}{2} \\cdot \\frac{\\rho^2 - x^2 -\ny^2}{\\rho\\sqrt{m^2+1} + x m -y}.$$ Hence, we proved the metric\nrelation$$\\frac{1}{4} \\left( \\frac{1}{R} + \\frac{1}{r} \\right)^2 =\n\\frac{4 \\rho^2}{(\\rho^2-x^2 - y^2)^2}.$$ By a straightforward\ncomputation, we can easily see that the Gaussian curvature of this\nmetric is $\\kappa _g=-1.$ Therefore this Riemannian metric\ngenerates the hyperbolic geometry on the disk.\n\\end{example}\n\nFor the next example, we apply Lemma \\ref{mimetic} to the\nfollowing.\n\n\\begin{proposition}Barbilian's metrization\nprocedure on $$K=\\{(x,0) \\in \\mathbb{R}^2 \/ x>0 \\} \\cup \\{(0,y)\\in\n\\mathbb{R}^2 \/y>0 \\},$$$$J= \\{(x,y)\\in \\mathbb{R}^2 \/ x>0, y>0\n\\}$$for the influence $f: K \\times J \\rightarrow \\mathbb{R}_+^*,$\ngiven by $f(M,A) = ||MA||$, yields the metric that at $(x_0,y_0)\n\\in J$ satisfies\\begin{equation}\\label{metrica_neriemanniana}ds^2\n=\\frac{(y_0m+x_0+(x_0+y_0) \\sqrt{m^2+1} )^2}{4x_0^2y_0^2\n(m^2+1)}(dx^2 + dy^2),\\end{equation}\n$m=\\frac{\\dot{y}}{\\dot{x}}|_{(x_0,y_0) },$ where the metric\n(\\ref{metrica_neriemanniana}) is a generalized Lagrange metric\nthat is not reducible to a Riemannian, Finslerian or Lagrangian\nmetric.\n\\end{proposition}\n\n{\\it Proof:} Denote as above $g_{AB}:K \\rightarrow\n\\mathbb{R}_+^*,$ given by $$g_{AB}(M) =\\frac{f(M,A)}{f(M,B)}=\n\\frac{||MA||}{||MB||}.$$First, we need to show that $g_{AB}$\nadmits maximum and minimum. Consider the points $A,B \\in J,$ $M\n\\in K,$ and denote by $A_1$ the foot of perpendicular from $A$ to\nthe $y-$axis and by $A_2$ the foot of perpendicular from $A$ on\nthe $x-$axis. Consider the inversion centered in $A$ and of power\n$||AA_1||^2.$ This inversion induces the correspondences $A_1\n\\rightarrow A_1,$ $A_2 \\rightarrow A_2'$ (such that $A_2' \\in\nOA_2$ and$||AA_2|| \\cdot ||AA_2'|| = ||AA_1||^2),$ $O \\rightarrow\nO'$ (such that $O' \\in AO$ and $A_1O' \\bot AO$), $B \\rightarrow\nB'$ such that $B' \\in AB$ and $||AB|| \\cdot ||AB'||= ||AA_1||^2$).\nThe positive part of the $y-$ axis is transformed in the arc of\ncircle $\\mathcal{C}_1$ of endpoints $A$ and $O',$ and it is part\nof the circle of diameter $AA';$ more precisely is the arc that\ncontains the point $A_1.$ The positive part of the $x-$ axis is\ntransformed in the arc of circle $\\mathcal{C}_2$ of endpoints $A$\nand $O',$ and it is part of the circle of diameter $AA_2'$, more\nprecisely the arc that contains the point $A_2.$ The inverse of a\npoint $M \\in K$ is part of the union of the two arcs described\nabove. Keeping in mind that\n\\begin{equation}\\label{formula}||B'M'|| = ||AA_1||^2 \\cdot\n\\frac{||BM||}{||AM|| \\cdot ||AB||}= \\frac{||AA_1||^2}{||AB||}\\cdot\n\\frac{||BM||}{||AM||},\\end{equation} we get that $||B'M'||$ is\nmaximum whenever $\\frac{||AM||}{||BM||}$ is minimum. Denote by\n$M_1'$ the point on $\\mathcal{C}_1 \\cup \\mathcal{C}_2$ for which\nis attained the maximum of the Euclidean distance $||B'M'||.$ The\nray $AM_1'$ intersects $K$ in $M_1$ for which $$m\n=\\frac{||AM_1||}{||BM_1||} = \\min_{M \\in K}\n\\frac{||AM||}{||BM||}.$$ From (\\ref{formula}) we deduce also that\nthere exists a point $M_2'$ for which $||B'M_2'||$ is the minimum\nfor $||B'M'||,$ when $M' \\in \\mathcal{C}_1 \\cup \\mathcal{C}_2.$\nThe inverse of $M_2'$ is $M_2$, obtained at the intersection\nbetween $AM_2'$ and $K$ and it has the property\n$$\\mathcal{M}=\\frac{||AM_2||}{||BM_2||}= \\max_{M \\in\nK}\\frac{||AM||}{||BM||}.$$ This allows us to conclude that the\nformula $d^B(A,B)= \\ln \\frac{\\mathcal{M}}{m}$ produces a Barbilian\ndistance in $J.$ Now we obtain the coefficients of the metric from\nLemma \\ref{mimetic}. Consider the arbitrary point $A(x_0,y_0) \\in\nJ$ and the line $(d)$ of equation $y-y_0=m(x-x_0).$ By Lemma\n\\ref{mimetic} there exist the circles $\\Gamma_1$ and $\\Gamma_2$\ntangent to the line $d$ in $A$ and tangent to $K.$ Denote by\n$O_1(x_1,y_1)$ the center of the circle $\\Gamma_1$ and by\n$O_2(x_2,y_2)$ the center of the circle $\\Gamma_2.$ To determine\nthe rays of the two circles described in Lemma \\ref{mimetic} (a)\nwe have the conditions\n\\begin{equation}\n\\label{prima}y_1 -y_0 = -\\frac{1}{m}(x_1 - x_0),  \\ \\ \\ x_1^2 =\n(x_1-x_0)^2 + (y_1-y_0)^2,\\end{equation} with $x_0 > x_1,$ and\n\\begin{equation}\n\\label{a doua} y_2-y_0= -\\frac{1}{m}(x_2 - x_0), \\ \\ \\ y_2^2 =\n(x_2-x_0)^2 + (y_2 - y_0)^2,\\end{equation} for $y_0 >y_1.$ From\n(\\ref{prima}) and (\\ref{a doua}), respectively, we obtain:\n\\begin{equation}R_1=x_1=\\frac{x_0\n\\sqrt{m^2+1}}{m+\\sqrt{m^2+1}}, \\ \\ \\ R_2=y_2=\\frac{y_0\n\\sqrt{m^2+1}}{1+\\sqrt{m^2+1}}.\\end{equation} Therefore, by\napplying Lemma \\ref{mimetic} the metric is expressed as in\n(\\ref{metrica_neriemanniana}). For the directions\n$m=\\frac{\\dot{y}}{\\dot{x}}$ with $\\dot{x} >0,$ the metric has the\ncoefficients\n\\begin{equation}g_{11}=g_{22}=\n\\frac{\\dot{x}(y \\cdot \\dot{y}+ x \\cdot \\dot{x} +(x+y)\n\\sqrt{\\dot{x}^2+\\dot{y}^2})^2}{4xy(\\dot{x}^2+\\dot{y}^2)}, \\ \\ \\\ng_{12}=g_{21}=0.\n\\end{equation}\nThis metric (see \\cite{M1994, MAB}) is a generalized Lagrange\nmetric, since the tensor expressed above is a $d$-tensor. To see\nthis, remark that the metric is $0-$homogeneous, and $\\det g =\n(g_{11})^2,$ therefore it is positive definite. According to\nsection 2.2 from \\cite{MAB}, the metric\n\\ref{metrica_neriemanniana} is reducible to a Lagrangian metric if\nand only if the Cartan tensor $C_{ijk}=\\frac{1}{2}\\frac{\\partial\ng_{ij}}{\\partial x^k}$ is totally symmetric (see \\cite{MAB},\nsection 4.1, Theorem 1.1.). The condition of symmetry reduces for\nthe metric (\\ref{metrica_neriemanniana}) to $$\\frac{\\partial\ng_{11}}{\\partial \\dot{y}}=\\frac{\\partial g_{12}}{\\partial\n\\dot{x}}.$$However, $\\frac{\\partial g_{12}}{\\partial\n\\dot{x}}\\equiv 0$ and $\\frac{\\partial g_{11}}{\\partial \\dot{y}}\n\\neq 0,$ which proves that the Cartan tensor is not totally\nsymmetric. Therefore, the metric (\\ref{metrica_neriemanniana}) is\nnot reducible to a Lagrangian metric. If the metric is not\nreducible to a Lagrangian metric, it is not reducible to either a\nFinslerian metric or a Riemannian metric. \\qed\n\n\\section{An Extension of Barbilian's Metrization\nProcedure}\n\nNow we present an extension of Barbilian's metrization procedure.\nOur motivation to produce this extension is the fact that in the\ncase when $K$ is a circle in the plane and $J$ is its interior, if\nwe remove one point $L$ from $K$, we can not apply the classical\nBarbilian's metrization procedure considering the influence of $K\n- \\{ L \\}$ over $J.$ Suppose that $K$ and $J$ are arbitrary sets\nand that they satisfy the {\\it general extremum requirement}, that\nis for any $A$ and $B$ in $J$ it exists  $\\sup g_{AB}(Q) <\n\\infty$, when $Q \\in K.$ As we have seen in the case of maximum,\nif there exists $\\sup_{P\\in K} g_{AB}(P) < \\infty$ then there\nexists $\\inf_{P\\in K} g_{AB}(P)$ and it equals $[\\sup_{P\\in K}\ng_{BA}(P)]^{-1}.$  We have the following (see also \\cite{WB2002},\np.10).\n\n\\begin{theorem} Suppose that $g$\nsatisfies the general extremum requirement. Then the function\n$d^s: J \\times J \\rightarrow \\mathbb{R}_+$ given by$$d^s(A,B) =\n\\ln \\frac{\\sup_{P\\in K} g_{AB}(P)}{\\inf_{P\\in K} g_{AB}(P)}$$ is a\nsemidistance on $J.$\n\\end{theorem}\n\n{\\it Proof:} We need to prove that: $d^s(A,B)+d^s(B,C) \\geq d^s\n(A,C).$ Then it is sufficient to show: $$\\frac{\\sup_{P\\in K}\ng_{AB}(P)}{\\inf_{P\\in K} g_{AB}(P)} \\cdot \\frac{\\sup_{Q\\in K}\ng_{BC}(Q)}{\\inf_{Q\\in K} g_{BC}(Q)} \\geq \\frac{\\sup_{R\\in K}\ng_{AC}(R)}{\\inf_{R\\in K} g_{AC}(R)}.$$ Denote by $\\alpha$ the left\nhand side term in the inequality above and remark that $$\\alpha\n\\geq \\frac{g_{AB}(P)}{g_{AB}(Q)} \\cdot \\frac{g_{BC}(Q)}{g_{BC}(P)}\n= \\frac{\\frac{f(P,A)}{f(P,B}}{\\frac{f(Q,A)}{f(Q,B)}}\\cdot\n\\frac{\\frac{f(P,B)}{f(P,C}}{\\frac{f(Q,B)}{f(Q,C)}}=\\frac{g_{AC}(P)}{g_{AC}(Q)},\n\\ \\ \\ \\ \\ \\forall P,Q \\in K.$$ This means $\\alpha \\cdot g_{AC}(Q)\n\\geq g_{AC}(P),$ for all $P,Q \\in K.$ Therefore, $$\\alpha \\cdot\ng_{AC}(Q) \\geq \\sup_{P \\in K} g_{AC}(P), \\ \\ \\ \\ \\ \\forall Q \\in\nK,$$ which yields $$\\alpha \\cdot \\inf_{Q \\in K} g_{AC}(Q) \\geq\n\\sup_{P \\in K} g_{AC}(P).$$ We obtain $$\\alpha \\geq \\frac{\\sup_{R\n\\in K} g_{AC}(R)}{\\inf_{R \\in K} g_{AC}(R)}.$$ \\qed\n\nThe authors address their thanks to Professors M. Anastasiei, D.\nE. Blair,  O. Dragi\\v{c}evi\\'{c} and John D. McCarthy for many\nuseful discussions on Barbilian's metrization procedure. The\nauthors thank also to the editor and the referees for their many\nuseful remarks that improved the content and the presentation of\nthis paper.\n\n{\\small\n\n\\bibliographystyle{amsplain}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}