diff --git "a/data_all_eng_slimpj/shuffled/split2/finalzzppeb" "b/data_all_eng_slimpj/shuffled/split2/finalzzppeb" new file mode 100644--- /dev/null +++ "b/data_all_eng_slimpj/shuffled/split2/finalzzppeb" @@ -0,0 +1,5 @@ +{"text":"\\section{Introduction and main result}\nBy a result of G Knieper \\cite{kni}, we know that the geodesic flow on\na compact rank 1 manifold $M$ admits a unique measure of \nmaximal entropy, concentrated on the open and dense set of\nregular vectors in $T^{1}M$.\nKnieper proved that this measure is\napproximated by probability measures supported on a finite number of\nregular closed geodesics.\nThis result generalize previous well known results for negatively curved\nmanifolds \\cite{ano} \\cite{bow} \\cite{bowe} \\cite{led} \\cite{mar}\n\\cite{marg} \\cite{cby}.\n\nIn this paper we consider the case of non constant potentials.\nWe obtain a formula expressing the topological\npressure as an exponential growth of the number of weighted regular closed\ngeodesics representing different free homotopy classes.\nAs a consequence we give an equidistribution result for weighted\nclosed regular geodesics to an equilibrium state. These results extend and\nstrengthens previous one by G Knieper (\\cite{kni} Proposition $6.4$)\nand M Pollicott \\cite{pol}. \nThe proof uses the Anosov's closing lemma for compact\nmanifolds of nonpositive curvature \\cite{BBS} and also a\nRiemannian formula for the topological pressure by G P Paternain \\cite{pat}. \n\nLet $M=X\/\\Gamma$ be a compact Riemannian manifold\nof nonpositive curvature where $X$ is the universal cover and $\\Gamma$\nis the group of deck transformations of $X$.\nThe rank of a vector $v\\in T^{1}M$ is the dimension of\nthe space of all parallel Jacobi fields along the geodesic defined by\n$v$. The rank of the manifold is the minimal rank of all tangeant\nvectors. We will assume that $M$ is a rank\n1 manifold (this includes manifolds of negative sectional curvature\nwhere all the geodesics are of rank 1). In fact,\nby a rigidity result of Ballmann\n\\cite{bal} and Burns-Spatzier \\cite{bs} ``most of'' compact manifolds of\nnonpositive curvature are rank 1.\nBy a regular vector (resp regular geodesic) we will mean a rank\n1 vector (resp a geodesic defined by a rank 1 vector). A geodesic is\ncalled hyperbolic if it is regular, extending thus the notion of\nhyperbolicity to rank $1$ manifolds. \nLet $\\mathcal{R}_{reg}$ be the open subset of $T^{1}M$ of regular vectors. It\nis dense in $T^{1}M$ if $M$ is of finite volume \\cite{ball}.\nLet $\\nu$ be the Knieper's measure of maximal entropy of the\ngeodesic flow of the rank 1 maniflod $M$. We have\n$\\nu(\\mathcal{R}_{reg})=1$ and the\ncomplement $\\mathcal{S}_{ing}$ of $\\mathcal{R}_{reg}$ is an invariant\nclosed subset of the unit tangent bundle. The growth of closed\ngeodesics in the ``singular part'' $\\mathcal{S}_{ing}$ can be\nexponential \\cite{gromov} as well as subexponential \\cite{kni}. \nIn this paper we concentrate on the regular set, but it will be\ninteresting to investigate the $\\mathcal{S}_{ing}$-part. \n\nTwo elements $\\alpha, \\beta \\in \\Gamma$ are equivalent if\nand only if there exists $n, m \\in \\intg$ and $\\gamma \\in \\Gamma$ such\nthat $\\alpha^{n}=\\gamma \\beta^{m} \\gamma^{-1}$. \nDenote by $[\\Gamma]$ the set of equivalence classes of elements in\n$\\Gamma$.\nClasses in $[\\Gamma]$\nare represented by elements in $\\Gamma$ which have a least period\n(primitive elements):\n\\[\n[\\alpha]=\\{\\gamma \\alpha_{0}^{m} \\gamma^{-1}: \\alpha_{0}\\in \\Gamma,\n\\ \\alpha_{0} \\ primitive, \\ \\gamma \\in \\Gamma \\}.\n\\]\nLet $x_{\\alpha}$ be the point in $X$ such that $d(x_{\\alpha}, \\alpha\nx_{\\alpha})=\\inf_{p\\in X}d(p,\\alpha p)$ ($M$ is compact). The axis\ntrougth $x_{\\alpha}$ and $\\alpha x_{\\alpha}$ projects onto\na closed geodesic in $M$ with prime period\n$d(x_{\\alpha}, \\alpha x_{\\alpha}):=l(\\alpha_{0})$.\nWe set $l([\\alpha]):=l(\\alpha_{0})$, i.e\n\\[\nl([\\alpha])=\\min \\{l(\\gamma): \\gamma \\in [\\alpha]\\}=l(\\alpha_{0}).\n\\]\nWe will denote by\n$\\Gamma_{hyp}\\subset \\Gamma$ the subset of those elements with hyperbolic\naxis. Then $[\\Gamma_{hyp}]$ is the set of conjugacy classes representing \ngeometrically distinct hyperbolic closed geodesics. Finally given a\nfunction $f$ on $T^{1}M$ the notation $\\int_{[\\alpha]}f$, $[\\alpha] \\in\n[\\Gamma_{hyp}]$, stands for the integral of $f$ along the unique closed\ngeodesic representing the class $[\\alpha]$. If this geodesic is given by \n $\\phi_{s}v_{[\\alpha]}, 0\\leq s \\leq l([\\alpha])$ for some\n$v_{[\\alpha]}\\in T^{1}M$, then, \n\\[\n\\int_{[\\alpha]}f:=\\int_{0}^{l([\\alpha])}f(\\phi_{s}v_{[\\alpha]})ds:=\n\\delta_{[\\alpha]} (f). \n\\]\nGiven a continuous function $f$ on $T^{1}M$, let $\\mu_{t}:=\\mu_{t}^{f}$ \nbe the flow invariant probability measures supported on a finite number of\n hyperbolic closed geodesics defined on continuous functions $\\omega$ by,\n\\[\n\\mu_{t}(\\omega):=\n\\frac{ \\sum_{([\\alpha] \\in \\Gamma_{hyp}:l([\\alpha])\\leq\n t)}e^{\\int_{\\alpha}f}\\delta_{[\\alpha]}(\\omega)} \n{ \\sum_{([\\alpha] \\in \\in \\Gamma_{hyp}: l([\\alpha])\\leq\n t)}e^{\\int_{\\alpha}f}}.\n\\]\nHere is the main result of the paper.\n\\begin{theorem} Let $M=X\/\\Gamma$ be a compact rank 1 manifold equipped with a\n $C^{\\infty}$ Riemannian metric and $f\\in\n \\mathcal{C}_{\\real}(T^{1}M)$. Then\n\\begin{enumerate}\n\\item \n\\begin{equation*}\n\\lim_{t\\rightarrow +\\infty}\\frac{1}{t}\n\\log \\sum_{[\\alpha] \\in [\\Gamma_{hyp}]: l([\\alpha])\\leq\nt}e^{\\int_{\\alpha}f}=P(f).\n\\end{equation*}\n\\item The accumulation points of $\\{\\mu_{t}\\}$ with respect to the \ntopology of weak convergence of measures, are equilibrium states of\nthe geodesic flow corresponding to \nthe potential $f$. Moreover, for any\nopen neighborhood $V$ in $\\mathcal{P}(T^{1}M)$ of the subset of\nequilibrium states $\\mathcal{P}_{e}(\\phi)$ we have,\n\\[\n\\lim_{t\\rightarrow +\\infty}\n\\frac{ \\sum_{([\\alpha] \\in \\Gamma_{hyp}:l([\\alpha])\\leq\n t, \\ \\delta_{[\\alpha]} \\in V)}e^{\\int_{\\alpha}f}} \n{ \\sum_{([\\alpha] \\in \\in \\Gamma_{hyp}: l([\\alpha])\\leq\n t)}e^{\\int_{[\\alpha]}f}}=1,\n\\]where the convergence is exponential.\n\\end{enumerate}\n\\end{theorem}\nIn the part $(2)$ of Theorem $1$, the condition $C^{\\infty}$ on the\nmetric is necessary since we need the upper-semicontinuity of the\nentropy map \\cite{new}. \n\nLet $d$ be the distance on $T^{1}M$ induced by the\nRiemannian metric of $M$.\nConsider the metric $d_{t}$ on $T^{1}M$, defined for all $t>0$ by\n\\[\nd_{t}(u,v):=\\sup_{0\\leq s \\leq t}d(\\phi^{s}(u), \\phi^{s}(v)).\n\\]\nFollowing \\cite{BBS} we denote by $P(t,\\epsilon)$ the maximal number\nof regular vectors $v\\in T^{1}M$ which are $\\epsilon$-separated in the metric\n$d_{t}$ and for which $\\phi^{t(v)}v=v$ for some $t(v) \\in [t,\nt+\\epsilon]$. Let $E(t,\\epsilon)$ be the set defined above with $\\#\nE(t,\\epsilon)= P(t,\\epsilon)$.\n \nThe following is the Lemma 5.6 from \\cite{BBS} for rank 1 manifolds\nand continuous potentials (see Lemma 1 $(5)$ below). \n\\begin{proposition} \nSet\n$\\int_{c_{v}}f:=\\int_{0}^{l(c_{v})}f(\\phi^{t}(v))dt$, where $c_{v}$ is\nthe closed geodesic defined by $v\\in E(t,\\epsilon)$ and $l(c_{v})$ is the\nperiod of $v$. Then, \n\\[\n\\lim_{\\epsilon \\rightarrow 0}\\liminf_{t\\rightarrow +\\infty}\\frac{1}{t}\n\\log \\sum_{v\\in E(t,\\epsilon)}e^{\\int_{c_{v}}f}=\n\\lim_{\\epsilon \\rightarrow 0}\\limsup_{t\\rightarrow +\\infty}\\frac{1}{t}\n\\log \\sum_{v\\in E(t,\\epsilon)}e^{\\int_{c_{v}}f}=P(f).\n\\]\n\\end{proposition} \n\\section{Proofs}\n\\subsection{Topological pressure}\nWe recall the notion of topological pressure \\cite{wal}.\nLet $t>0$ and $\\epsilon >0$. A subset $E\\subset T^{1}M$ is a\n$(t,\\epsilon)$-separated if $d_{t}(u,v)>\\epsilon$ for $u \\ne v \\in E$.\nSet\n\\[\nr(f; t,\\epsilon):=\\sup_{E}\\sum_{\\theta \\in\nE}e^{\\int_{0}^{t}f(\\phi_{t}(\\theta))dt} \n\\] \nwhere $\\sup$ is over all $(t,\\epsilon)$-separated subsets $E$; and \n\\[\nr(f;\\epsilon):=\\limsup_{t\\rightarrow \\infty}\\frac{1}{t}\n\\log r(f; t,\\epsilon).\n\\]\nThen the topological pressure of the geodesic flow corresponding to the\npotential de $f$ is the number, \n\\begin{equation}\nP(f)=\\lim_{\\epsilon \\rightarrow 0}r(f;\\epsilon).\n\\end{equation}\nThe topological entropy $h_{top}$ is $h_{top}=P(0)$.\nWe denote by $\\mathcal{P}(T^{1}M)$ the set of all probability mesures on\n$T^{1}M$ with the weak topology of mesures, and let $\\mathcal{P}(\\phi)$ be\nthe subset of invariant probability mesures of the flow.\nThe entropy of a probability measure $m$ is denoted $h(\\mu)$ \\cite{wal}.\nAll these objects satisfy the following variational principle \\cite{wal}\n\\begin{equation}\nP(f)=\\sup_{\\mu\\in \\mathcal{P}(\\phi)}\\left (h(\\mu)+\\int_{T^{1}M}fd\\mu\n\\right ).\n\\end{equation}\nAn equilibrium state $\\mu_{f}$ satisfies,\n\\begin{equation}\nh(\\mu_{f})+\\int_{T^{1}M}fd\\mu_{f}=P(f).\n\\end{equation}\nWhen the Riemannian metric of the manifold $M$ is $C^{\\infty}$ then by\na result of Newhouse \\cite{new} the entropy map $m\\rightarrow h(m)$ is\nupper semicontinuous and then $h_{top}<\\infty$. Consequently, the\nset $\\mathcal{P}_{e}(f)$ of equilibrium states\nis a non empty closed and convex subset of $\\mathcal{P}(\\phi)$.\n\\subsection{Proof of Theorem $1$ $(1)$}\nLet $\\nu$ be the Knieper's measure. Let $N(t, \\epsilon, 1-\\delta,\n\\nu)$ be the minimal number of $\\epsilon$-balls in the metric $d_{t}$\nwhich cover a set of measure at least $1-\\delta$. Since $\\nu$ is\nthe unique measure of maximal entropy, we can apply Lemma 5.6 in\n\\cite{BBS} to this measure. \n\\begin{lemma}[\\cite{BBS}] There exists $\\delta >0$ such that for all\n$\\epsilon >0$, there exists $t_{1}>0$ such that \n\\begin{equation}\nP(t,\\epsilon)\\geq N(t, \\epsilon, 1-\\delta, \\nu)\n\\end{equation}\nfor any $t\\geq t_{1}$. In particular, we have\n\\begin{equation}\n\\lim_{\\epsilon \\rightarrow 0}\\liminf_{t\\rightarrow +\\infty}\\frac{\\ln\n P(t,\\epsilon)}{t}= \\lim_{\\epsilon \\rightarrow 0}\\limsup_{t\\rightarrow\n +\\infty}\\frac{\\ln P(t,\\epsilon)}{t}=h_{top}.\n\\end{equation}\n\\end{lemma}\nWe fix $\\epsilon >0$. Recall that $E(t,\\epsilon)$ is the maximal set\ndefined above with $\\# E(t,\\epsilon)=P(t,\\epsilon)$.\nFrom Lemma 1 $(4)$, if $N(t,\\epsilon, 1-\\delta, \\nu)$ is the minimal\nnumber of $\\epsilon$-balls $B_{t}(v_{i},\\epsilon)$ in the metric $d_{t}$,\nwhich cover the whole space $T^{1}M$, then\n$P(t,\\epsilon)=N(t,\\epsilon, 1-\\delta,\\nu)$ for $t$ sufficiently large\n(since each $\\epsilon$-ball $B_{t}(v_{i},\\epsilon)$ contains a unique\npoint from $E(t,\\epsilon)$). \n\nNow, it suffices to prove Theorem $1$ ($1$) for Lipschitz functions $f$.\nSuppose then $f$ Lipschitz and let $lip(f)$ be it's Lipschitz constant.\nConsider a $(2\\epsilon,t)$-separated set $E_{1}$ in\n$T^{1}M$. Thus, two distinct vectors in $E_{1}$ lies in two distinct\n$\\epsilon$-balls above, so that $\\# E_{1}\\leq P(t,\\epsilon)$. \nFor each $\\theta \\in E_{1}$, we associate the unique point $v_{\\theta} \\in\nE(t, \\epsilon)$ such that $d_{t}(\\theta,v_{\\theta} )\\leq\n\\epsilon$. Let $\\tau_{\\theta}$ the regular closed geodesic\ncorresponding to the regular periodic vector $v_{\\theta}$\n($\\dot{\\tau}_{\\theta}(0)=v_{\\theta}$), with period $l(\\tau_{\\theta})\\in [t,\n t+\\epsilon]$, and \n$[\\tau_{\\theta}]$ the corresponding free homotopy class.\nThere exists a constant $C>lip(f)$ such that,\n\\begin{eqnarray*}\n&&\\sum_{\\theta \\in E_{1}}e^{\\int_{0}^{t} f(\\phi_{s}(\\theta))ds}\\\\\n&\\leq & e^{lip(f)\\epsilon t} \\sum_{[\\tau_{\\theta}]: \\theta\n\\in E_{1}, t0$ we have,\n\\begin{equation*}\nP(f)=\\lim_{t\\rightarrow \\infty}\n\\frac{1}{t}\n\\log \\int_{M\\times M}\n\\left (\\sum_{\\gamma_{xy}: t-\\delta 0$ and $(x,y)\\in M\\times M$ we consider the subset\nof $T^{1}M$ defined by,\n\\[\nE_{xy}:=\\{\\dot{\\gamma}_{xy}(0): t-\\delta < l(\\gamma_{xy}) \\leq t\\}.\n\\]\nIt is finite for almost all $(x,y)\\in M\\times M$ \\cite{patt}. As\nconsequence of the nonpositive curvature, the rank 1 manifold $M$ has no\nconjugate points.\nThus, there exists a constant $\\epsilon_{0}$, depending only on $M$,\nsuch that $E_{xy}$ is $(2\\epsilon,t)$-separated for $\\epsilon\n<\\epsilon_{0}$. To see this, it suffices to lift every thing to the\nuniversal cover of $M$ and use (\\cite{pat} p135) and (\\cite{has} p375). \nThus, the preceeding arguments applied to $ E_{1}=E_{xy}$ give,\n\\begin{eqnarray*}\n&& \\liminf_{t\\rightarrow \\infty}\\frac{1}{t} \n\\log \\int_{M\\times M}\\left ( \\sum_{\\theta \\in E_{xy}}e^{\\int_{0}^{t}\n f(\\phi^{s}(\\theta))ds}\n\\right )dxdy\\\\\n&\\leq& \\liminf_{t\\rightarrow \\infty}\\frac{1}{t} \n\\log \\left (\\sum_{[\\alpha] \\in \\Gamma_{hyp}:l([\\alpha])\\leq t}\ne^{\\int_{[\\alpha]} f}\\right ).\n\\end{eqnarray*}\nBut by Theorem 2, the left hand side of this inequality is a limit and\nis equal to $P(f)$, which completes the proof.\n\\begin{remarkk}\nFind a proof which did not appeal to Paternain's formula in Theorem 2 !\n\\end{remarkk}\n\\subsection{Proof of Proposition 2}\nThe proof of Proposition 2 follows from the above arguments. \n\\subsection{Proof of Theorem $1$ $(2)$}\nConsider the following functional which measures the\n``distance'' of an invariant measure $m$ to the set of equilibrium states,\n\\[\n\\rho(m)=P(f)-\\left (h(m)+\\int fdm \\right ).\n\\]\nSet $\\rho(E):=\\inf (\\rho(m): m\\in E)$ for $E \\subset\n\\mathcal{P}(\\phi)$ and,\n\\[\n[\\Gamma_{hyp}](t):=\\{[\\alpha] \\in [\\Gamma_{hyp}]: l([\\alpha])\\leq t\\}.\n\\]\n\\begin{lemma} Let $M$ be a compact smooth rank 1 manifold and\n $f$ a continuous potential on $T^{1}M$.\nThen, for any closed subset $K$ of $\\mathcal{P}(\\phi)$ we have,\n\\[\n\\limsup_{t\\rightarrow +\\infty }\\frac{1}{t}\\log\n \\frac{\\sum_{([\\alpha] \\in [\\Gamma_{hyp}](t): \\delta_{[\\alpha]} \\in K)}\n e^{\\int_{[\\alpha]} f}} \n{\\sum_{[\\alpha] \\in [\\Gamma_{hyp}](t)}\n e^{\\int_{[\\alpha]} f}} \n\\leq -\\rho(K).\n\\]\n\\end{lemma}\nWe leave the proof of this lemma for later and finish the proof of\nTheorem $1$. First, let $V$ be an open neighborhood of\n$\\mathcal{P}_{e}(f)$ and set $K=\\mathcal{P}(\\phi)\\backslash V$. The\nset $K$ is compact and $\\rho(K)>0$. \nFor $t$ sufficiently large we have by Lemma $3$, \n\\[\n1\\geq \\frac{\\sum_{([\\alpha] \\in [\\Gamma_{hyp}](t): \\delta_{[\\alpha]} \\in V)}\n e^{\\int_{[\\alpha]} f}} \n{\\sum_{[\\alpha] \\in [\\Gamma_{hyp}](t)}\n e^{\\int_{[\\alpha]} f}}\\geq 1-e^{-t\\rho(K)}.\n\\]\nThis proves the second assertion in part $(2)$ of the Theorem $1$. \n\n\\subsubsection{Accumulation measures of $\\mu_{t}$}\nThe proof of the fact that the accumulation measures of $\\mu_{t}$ are in\n$\\mathcal{P}_{e}(\\phi)$ follows \\cite{am}.\n\nWe endow $\\mathcal{P}(T^{1}M)$ with a distance $d$ compatible with the weak\nstar topology: take a countable base $\\{g_{1}, g_{2}, \\cdots\\}$ of the\nseparable space \n$C_{\\real}(T^{1}M)$, where $\\|g_{k}\\|=1$ for all $k$, and set:\n\\[\nd(m, m'):=\\sum_{k=1}^{\\infty}2^{-k}\\left |\\int g_{k}dm - \\int g_{k}dm'\n\\right |.\n\\] \n \nLet $V\\subset \\mathcal{P}(\\phi)$ be a convex open \nneighborhood of $\\mathcal{P}_{e}(f)$ and $\\epsilon >0$.\nWe consider a finite open cover $(B_{i}(\\epsilon))_{i\\leq N}$ of\n$\\mathcal{P}_{e}(f)$ by balls of diameter $\\epsilon$ all contained in $V$. \nDecompose the set $U:=\\cup_{i=1}^{N}B_{i}(\\epsilon)$ as follows,\n\\[\nU=\\cup_{j=1}^{N'}U_{j}^{\\epsilon},\n\\]where the sets $U_{j}^{\\epsilon}$ are disjoints (not necessarily\nopen ) and contained in one\nof the balls $(B_{i}(\\epsilon))_{i\\leq N}$. We have\n\\[\n\\mathcal{P}_{e}(f) \\subset U \\subset V.\n\\] \nWe fix in each $U_{j}^{\\epsilon}$ an invariant probability measure $m_{j}$,\n$j\\leq N'$, and let $m_{0}$ be an invariant probability measure\ndistinct from the above ones; for example take $m_{0} \\in V\\backslash U$.\nSet for convenience,\n\\begin{equation}\n\\nu_{t}(E):= \\frac{\\sum_{([\\alpha] \\in [\\Gamma_{hyp}](t):\n \\delta_{[\\alpha]} \\in E)} e^{\\int_{[\\alpha]} f}} \n{\\sum_{[\\alpha] \\in [\\Gamma_{hyp}](t)}\n e^{\\int_{[\\alpha]} f}},\\\n E\\subset \\mathcal{P}(\\phi)\n\\end{equation}\nand define,\n\\begin{equation}\n\\beta_{t}=\n\\sum_{j=1}^{N'}\\nu_{t}(U_{j}^{\\epsilon})m_{j}+\n(1-\\nu_{t}(U))m_{0}. \n\\end{equation}\nWe have $\\sum_{j=1}^{N'}\\nu_{t}(U_{j}^{\\epsilon})=\\nu_{t}(U)$.\nThe probability measure $\\beta_{t}$ lies in $V$ since it is a convex\ncombination of elements in the convex set $V$. Thus\n\\[\nd(\\mu_{t}, V)\\leq d(\\mu_{t},\\beta_{t}). \n\\]\nWe are going to show that\n\\[\nd(\\mu_{t},\\beta_{t})\\leq \\epsilon \\nu_{t}(U)+3\\nu_{t}(U^{c}),\n\\]\nwhere $U^{c}=\\mathcal{P}(SM)\\backslash U$.\n\nConsider the measures $\\mu_{t,V}$ on $SM$ defined by,\n\\[\n\\mu_{t,V}:=\\frac{\\sum_{([\\alpha] \\in [\\Gamma_{hyp}](t):\n \\delta_{[\\alpha]} \\in V)} \n e^{\\int_{[\\alpha]} f}\\delta_{[\\alpha]}} \n{\\sum_{[\\alpha]\\in [\\Gamma_{hyp}](t)}\n e^{\\int_{[\\alpha]} f}}.\n\\]\nBy definition of $\\mu_{t}$ and $\\mu_{t,V}$ and\nthe fact that $U\\subset V$,\n\\[\n\\sum_{k\\geq 1}2^{-k}|\\mu_{t}(g_{k})-\\mu_{t,V}(g_{k})|\\leq\n\\nu_{t}(U^{c}).\n\\]\nIt remains to show that \n\\[\n\\sum_{k\\geq 1}2^{-k}|\\mu_{t,V}(g_{k})-\\beta_{t}(g_{k})| \\leq \\epsilon\n\\nu_{t}(U)+\\nu_{t}(U^{c}).\n\\] \nWe have for all $k\\geq 1$,\n\\[\n|\\mu_{t,V}(g_{k})-\\beta_{t}(g_{k})|\\leq A+B+C\n\\]where,\n\\[\nA=\n\\frac{\\sum_{j=1}^{N'}\\sum_{[\\alpha] \\in [\\Gamma_{hyp}](t):\n\\delta_{[\\alpha]}\\in U_{j}^{\\epsilon}}\ne^{\\int_{[\\alpha]}f}|\\delta_{[\\alpha]}(g_{k})-m_{j}(g_{k})|}\n{\\sum_{[\\alpha] \\in [\\Gamma_{hyp}](t)}e^{\\int_{[\\alpha}] f}},\n\\]\n\\[\nB=\\frac{\\sum_{[\\alpha] \\in [\\Gamma_{hyp}](t):\n\\delta_{[\\alpha]}\\in V\\backslash\nU}e^{\\int_{[\\alpha]}f}\\delta_{[\\alpha]}(g_{k})} \n{\\sum_{[\\alpha] \\in [\\Gamma_{hyp}](t)}e^{\\int_{[\\alpha}] f}},\n\\]\n\\[\nC=|(1-\\nu_{t}(U))m_{0}(g_{k})|.\n\\]\nThus, since we have for all $k\\geq 1$, $\\|g_{k}\\|=1$, by definition\nof $\\nu_{t}$ we get,\n\\begin{eqnarray*}\n&&\\sum_{k\\geq 1}2^{-k}|\\mu_{t,V}(g_{k})-\\beta_{t}(g_{k})|\\\\\n&\\leq& \\epsilon\n \\sum_{j=1}^{N'}\\nu_{t}(U_{j}^{\\epsilon})+\\nu_{t}(U^{c})+\n(1-\\nu_{t}(U))\\\\\n&=& \\epsilon \\nu_{t}(U)+\\nu_{t}(U^{c}).\n\\end{eqnarray*}\nFinally we have obtained that\n\\[\nd(\\mu_{t},\\beta_{t})\\leq \\epsilon\n\\nu_{t}(U)+\\nu_{t}(U^{c}). \n\\]\nThis implies the desired inequality,\n\\[\nd(\\mu_{t},V)\\leq \\epsilon\n\\nu_{t}(U)+3\\nu_{t}(U^{c}). \n\\]\nThe set $U^{c}$ is closed, so we have $\\lim_{t\\rightarrow\n\\infty}\\nu_{t}(U)=1$. Since $\\epsilon$ is arbitrary, we\nconclude that $\\limsup_{t\\rightarrow\n\\infty}d(\\mu_{t},V)=0$. The neighborhood $V$ of\n$\\mathcal{P}_{e}(f)$ being arbitrary, this implies that all limit\nmeasures of $\\mu_{t}$ are contained in $\\mathcal{P}_{e}(f)$. In\nparticular, if $\\mathcal{P}_{e}(f)$ is reduced to one measure $\\mu$, this\nshows that $\\mu_{t}$ converges to $\\mu$.\n\n\\subsection{Proof of Lemma $3$}\nWe follow \\cite{pol}.\nThe functional $\\rho$ is lower semicontinuous (since $h$ is upper\nsemicontinuous) and $\\rho \\geq 0$. \nSet for any continuous function $\\omega$ on $T^{1}M$,\n\\begin{equation}\nQ_{f}(\\omega):=P(f+\\omega)-P(f).\n\\end{equation}\nThe fact that $Q_{f}$ is a convex and continuous is a consequence of\nthe same properties for $P$.\nUsing the variational principle, it is not difficult to see that\n\\[\nQ_{f}(\\omega)=\\sup_{\\mu \\in \\mathcal{P}(\\phi)}\\left (\\int \\omega\n d\\mu-\\rho(\\mu)\\right ).\n\\]\nBy duality we have for any invariant probabilit\u00e9 measure $m$,\n\\[\n\\rho(m)=\\sup_{\\omega \\ continuous}\\left (\\int \\omega\n dm-Q_{f}(\\omega)\\right ).\n\\]\nWith the notations introduced above, we have to prove that\n\\[\n\\limsup_{t\\rightarrow +\\infty}\\frac{1}{t}\\log \\nu_{t}(K)\\leq -\\rho(K).\n\\]\nLet $\\epsilon >0$. There exists a finite number of continuous\nfunctions $\\omega_{1}, \\cdots, \\omega_{l}$ such that $K\\subset\n\\cup_{i=1}^{l}K_{i}$, where \n\\[\nK_{i}=\\{m\\in \\mathcal{P}(\\phi): \\int\n\\omega_{i}dm-Q(\\omega_{i})>\\rho(K)-\\epsilon\\}. \n\\]\nWe have $\\nu_{t}(K) \\leq \\sum_{i=1}^{l}\\nu_{t}(K_{i})$ where\n\\[\n\\nu_{t}(K_{i})=\\frac{\\sum_{([\\alpha] \\in [\\Gamma_{hyp}](t):\n \\delta_{[\\alpha}] \\in K_{i})} e^{\\int_{[\\alpha]} f}}\n{\\sum_{[\\alpha] \\in [\\Gamma_{hyp}](t)}e^{\\int_{[\\alpha]} f}}.\n\\]\nThen,\n\\begin{eqnarray*}\n&&\\sum_{[\\alpha] \\in [\\Gamma_{hyp}](t):\n\\delta_{[\\alpha]} \\in K_{i}} e^{\\int_{[\\alpha]} f}\\\\\n&\\leq&\\sum_{[\\alpha] \\in [\\Gamma_{hyp}](t):\n\\delta_{[\\alpha]} \\in K_{i}} e^{\\int_{[\\alpha]} f}\ne^{l([\\alpha])(\\int \\omega_{i}d\\delta_{[\\alpha]}-\nQ(\\omega_{i})-(\\rho(K)-\\epsilon))}.\n\\end{eqnarray*}\nSet $C:=\\sum_{i\\leq l}\\sup(1, e^{-\\delta(-Q(\\omega_{i})-(\\rho(K)-\\epsilon))})$.\nThus, by taking into account the sign of $-Q(\\omega_{i})-(\\rho(K)-\\epsilon)$,\n\\begin{eqnarray*}\n&& \\sum_{\\alpha \\in [\\Gamma_{hyp}](t):\n\\delta_{[\\alpha]} \\in K_{i}} e^{\\int_{[\\alpha]} f}\\\\\n&\\leq& \\sum_{[\\alpha] \\in [\\Gamma_{hyp}](t):\n\\delta_{[\\alpha]} \\in K_{i}} e^{\\int_{[\\alpha]} (f+\\omega_{i})}\ne^{l([\\alpha])(-Q(\\omega_{i})-(\\rho(K)-\\epsilon))}\\\\\n&\\leq& C e^{t\\left ( \n-Q(\\omega_{i})-(\\rho(K)-\\epsilon)\\right )}\n\\sum_{[\\alpha] \\in [\\Gamma_{hyp}](t):\n\\delta_{[\\alpha]} \\in K_{i}} e^{\\int_{[\\alpha]} (f+\\omega_{i})}.\n\\end{eqnarray*}\nFor $t$ sufficiently large, it follows from Theorem $1$ $(1)$,\n\\begin{eqnarray*}\n\\nu_{t}(K)& \\leq& \nC\\sum_{i=1}^{l}e^{t(P(f+\\omega_{i})+\\epsilon)}e^{-t(P(f)-\\epsilon)}\ne^{t(-Q(\\omega_{i})-(\\rho(K)-\\epsilon)}\\\\\n&=&Cl e^{t(-\\rho(K)+3\\epsilon)}.\n\\end{eqnarray*}\nTake the logarithme, divide by $t$ and take the $\\limsup$,\n\\[\n\\limsup_{t\\rightarrow \\infty}\\frac{1}{t}\\log \\nu_{t}(K) \\leq\n-\\rho(K)+3\\epsilon. \n\\]\n$\\epsilon$ being arbitrary, this proves Lemma $3$.\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Introduction}\n\nBoolean algebra for Boolean parameters (often represented by binomial variables 0, 1) is a field of mathematics \\citep{givant_introduction_2009} and has been applied to other fields, e.g., circuit of computer science, cryptography, and medicine \\citep{wu_boolean_2016}.\nIn these applications, Boolean function $f:\\{0,1\\}^n\\rightarrow\\{0,1\\}$ is constructed by ``$not$,'' ``$and$,'' ``$or$,'' and \"parentheses.\" In medicine, 0 and 1 are used to represent medical test results and whether a patient has a disease or a symptom. One of the most useful applications of binominal variables in medicine is a contingency table. Contingency tables enable us to discuss the usefulness of a medical test and diagnostic performance. However, it is difficult to apply a contingency table to a situation with multivariables. When we use $N$ binomial variables, the number of categories is $2^N$. The number of possible categories increases exponentially, making it difficult to model interactions of variables. Moreover, taking the interaction of variables into consideration, we cannot get a large enough sample size for each category. We hypothesize that randomly assigning Boolean operators and focusing on frequencies of Boolean operators could explain the outcome correctly, obtain the tendencies of operators, and overcome difficulties in analyzing large numbers of variables and categories. The aims of this paper are introducing a method called the Boolean Monte Carlo Method (BMCM) to obtain tendencies of Boolean operators and expanding 2 by 2 contingency tables for use in multivariate situations.\n\n\n\\section{Method}\n\nThis method was composed of three steps.\n\nIn the first step, we defined 4 kinds of null data as follows: 1) all explanatory variables were 1 and the objective variable was 1, 2) all explanatory variables were 0 and the objective variable was 0, 3) all explanatory variables were 1 and the objective variable was 0 and 4) all explanatory variables were 0 and the objective variable was 1. Then we created data that only contained null data and performed a chi-square test (Fisher's exact test). This was done because results from a Boolean operation for null data are trivial (Figure 1 (a)). For example, when all explanatory variables were 0, the result was 0 whether ``$and$'' or ``$or$'' was used for each operation. When performing a chi-square test for a 2 by 2 contingency table that includes null data with trivial results, we might underestimate the trends of data. Meanwhile, we could use null data to see a rough trend. When explanatory variables are all 0 (1), objective variables should tend to be 0 (1). If there was no such trend, this method would make no sense, and we should change the variables or hypothesis.\n\nIn the second step, we constructed a model and analyzed the operator trends. We constructed a model and randomly allocated operators between explanatory variables. In constructing the model, we adjusted orders of explanatory variables with parentheses. We performed operations and checked whether the randomly assigned operators could explain an objective variable. If the operator cannot explain the objective variable, we call this type of operator \"unfaithful.\" After we examined all data, we analyzed each operator that was grouped as \"faithful\" with a chi-square binomial test. As operators were assigned by the same probability, 1\/2, we could detect a tendency by examining the frequency of ``$and$'' or ``$or$.'' After the second step, we determined proper operators (Appendix A).\n\nIn the third step, proper operators from the second step were used. We applied these operators to the original data and made a 2 by 2 contingency table. Finally, we performed a chi-square test. In Figure 1 (b), the summary of these steps is shown. In this paper, we call this the Boolean Monte Carlo Method (BMCM).\n\n\\begin{figure}\n\\centering\n\\subfloat[Null data such as 1) and 2) were satisfied whether we used ``$and$'' or ``$or$'' for each operation.Each combination of operators was obtained by probability $1\/2^n$ and trend might be hidden. Null data such as 3) and 4) were not satisfied whether we used ``$and$'' or ``$or$'' for each operation.]\n\\resizebox*{5cm}{!}{\\includegraphics{nulldata.eps}}}\\hspace{5pt}\n\\subfloat[Three steps of boolean monte calro method. 1st step, null data was analyzed. 2nd step, data excluded null data was analyzed. 3rd step, analysis for original data was performed.]\n\\resizebox*{5cm}{!}{\\includegraphics{bmc_figure.eps}}}\n\\caption{Null data explanation and scheme of bmcm } \\label{Figure1}\n\\end{figure}\n\n\\begin{table}\n\\tbl{Categorization of Boolean function calculation and outcome. When the Boolean function calculation was equal with outcome, we categorized it as \"faithful.\" When the Boolean function calculation was not equal with outcome, we categorized it as \"unfaithful.\"}\n{\\begin{tabular}{ccc} \\toprule\n& $x_O=1$ & $x_O=0$ \\\\ \\midrule\n$f(x_1,\\cdots,x_n)=1$& faithful & unfaithful \\\\\n$f(x_1,\\cdots,x_n)=0$& unfaithful & faithful \\\\ \\bottomrule\n\\label{tb:tab1}\n\\end{tabular}}\n\\label{Table1}\n\\end{table}\n\n\n\\section{Example}\n\nIn this sectoin, we applied the BMCM to a sample data set. The data contained 1,000 observations for three variables $x_1, x_2, x_3$ and outcome $x_O$. In this data, $x_1$ determined $x_O$, and $x_2$ and $x_3$ were determined randomly, i.e., if $x_1=1$ then $x_O=1$ and if $x_1=0$ then $x_O=0$. The half of $x_1$ and half of $x_O$ were 1 (Figure 2).\n\n\\begin{figure}\n\\centering\n\\resizebox*{5cm}{!}{\\includegraphics{random23.eps}}\\hspace{5pt}\n\\caption{$x_1$ determined $x_O$, and $x_2$ and $x_3$ were determined randomly. } \\label{Figure1}\n\\end{figure}\n\n\n\\subsection{Null data analysis}\n\nTable 2 shows the contingency table for null data. Fisher's exact test showed that there was a significant difference in the proportions of $x_O$ between $x_1=x_2=x_3=1$ and $x_1=x_2=x_3=0$ $(\\chi^2=2.4\\times 10^2, p=1.1\\times 10^{-53})$.\n\n\n\\begin{table}\n\\tbl{Null data analysis for data in which $x_1$ associated with $x_O$.}\n{\\begin{tabular}{lrr} \\toprule\n& $x_O=1$ & $x_O=0$ \\\\ \\midrule\nall 1& 120 & 0 \\\\\nall 0 & 0 & 122 \\\\ \\bottomrule\n\\end{tabular}}\n\\label{Table2}\n\\end{table}\n\n\n\\vspace{10pt}\n\\noindent Model: $x_1$\\hspace{2pt}\\fbox{1}\\hspace{2pt}$x_2$\\hspace{2pt}\\fbox{2}\\hspace{2pt}$x_3=x_O$\n\nA chi square binomial test for faithful data showed that in operator \\hspace{2pt}\\fbox{1}\\hspace{2pt} ``$or$'' had a higher probability than ``$and$'' $(\\chi^2=41, p=1.6\\times 10^{-10})$ and in operator \\hspace{2pt}\\fbox{2}\\hspace{2pt} ``$and$'' had a higher probability than ``$or$'' $(\\chi^2=42, p=8.5\\times 10^{-11})$. Similarly, a chi-square binomial test for unfaithful data showed that in operator \\hspace{2pt}\\fbox{1}\\hspace{2pt} ``$and$'' had a higher probability than ``$or$'' $(\\chi^2=55, p=1.0\\times 10^{-13})$ and in operator \\hspace{2pt}\\fbox{2}\\hspace{2pt} ``$or$'' had a higher probability than ``$and$'' $(\\chi^2=57, p=4.4\\times 10^{-14})$.\n\n\\vspace{10pt}\n\\noindent Model: $x_1$\\hspace{2pt}\\fbox{1}\\hspace{2pt}$x_3$\\hspace{2pt}\\fbox{2}\\hspace{2pt}$x_2=x_O$\n\nA chi square binomial test for faithful data showed that in operator \\hspace{2pt}\\fbox{1}\\hspace{2pt} ``$or$'' had a higher probability than ``$and$'' $(\\chi^2=34, p=6.4\\times 10^{-9})$ and in operator \\hspace{2pt}\\fbox{2}\\hspace{2pt} ``$and$'' had a higher probability than ``$or$'' $(\\chi^2=35, p=3.6\\times 10^{-9})$. Similarly, a chi-square binomial test for unfaithful data showed that in operator \\hspace{2pt}\\fbox{1}\\hspace{2pt} ``$and$'' had a higher probability than ``$or$'' $(\\chi^2=47, p=7.0\\times 10^{-12})$ and in operator \\hspace{2pt}\\fbox{2}\\hspace{2pt} ``$or$'' had a higher probability than ``$and$'' $(\\chi^2=49, p=3.2\\times 10^{-12})$.\n\n\\begin{table}\n\\tbl{Contingency table for 3rd step.}\n{\\begin{tabular}{lrr} \\toprule\n & $x_O=1$ & $x_O=0$ \\\\ \\midrule\n $f(x_1, x_2, x_3) = 1$& 500 & 131 \\\\\n $f(x_1, x_2, x_3) = 0$ & 0 & 369 \\\\ \\bottomrule\n \\end{tabular}}\n\\label{Table2}\n\\end{table}\n\n\\vspace{10pt}\nFor the third step, we analyzed the original data when the operator in \\hspace{2pt}\\fbox{1}\\hspace{2pt} is ``$or$'' and the operator in \\hspace{2pt}\\fbox{2}\\hspace{2pt} is ``$and$.\" Table 3 shows the result of the analysis of the original dataset, including null data and other data. Fisher's exact test showed that there was a significant difference in the proportions of $x_O$ between $f(x_1,x_2,x_3)=1$ and $f(x_1,x_2,x_3)=0 (\\chi^2=5.8\\times 10^2, p=1.7\\times 10^{-128})$.\n\n\n\n\n\n\n\\section{Discussion}\nIn this study, we proposed a new method (BMCM) to model interactions of binomial variables by assigning Boolean operators, and to expand the 2 by 2 contingency table to handle multivariate cases. We applied the BMCM to data of which variables and outcome were randomly determined. This analysis showed that there was no significant result (Appendix).\nWe applied the BMCM to data that was dependent on one variable. This data had null data, which was significant, and some operators could be determined. In the model $x_1$\\hspace{2pt}\\fbox{1}\\hspace{2pt}$x_2$\\hspace{2pt}\\fbox{2}\\hspace{2pt}$x_3=x_O$, \\hspace{2pt}\\fbox{1}\\hspace{2pt} tended to be ``$or$'' and \\hspace{2pt}\\fbox{2}\\hspace{2pt} tended to be ``$and$.\" When \\hspace{2pt}\\fbox{1}\\hspace{2pt} was ``$or$'' and \\hspace{2pt}\\fbox{2}\\hspace{2pt} was ``$and$,\" the calculation between $x_2$ and $x_3$ was performed first, and then the calculation between $x_1$ and the result of $x_2$\\hspace{2pt}\\fbox{2}\\hspace{2pt}$x_3$ was performed. This model with operators (\\hspace{2pt}\\fbox{1}\\hspace{2pt}$=or$, \\hspace{2pt}\\fbox{2}\\hspace{2pt}$=and$) might reflect the fact that $x_1$ determined $x_O$, and $x_2$ and $x_3$ were assigned randomly. We could interpret the model $x_1$\\hspace{2pt}\\fbox{1}\\hspace{2pt}$x_3$\\hspace{2pt}\\fbox{2}\\hspace{2pt}$x_2=x_O$, in the same way because there was symmetry between $x_2$ and $x_3$.\nApplications of Boolean functions have been attempted in medicine. Previous studies applied this method mainly for the gene regulatory network (GRN). In these studies, to construct GRN models, operators between variables should be determined using methods such as the Bayesian approach \\citep{lin_logic_2014,zhou_gene_2004}, Markov chain approach \\citep{xiao_tutorial_2009}, and satisfiability problem solver (SAT solver) approach \\citep{moskewicz_chaff:_nodate,lin_application_2012}. These studies construct models of mutual interaction of genes by operators. These mutual interactions often can be interpreted as a GRN. A hypothesis of constructing a GRN randomly was also reported. Kauffman\\citep{kauffman_metabolic_1969} studied an approach for a GRN that is randomly connected. Pal R et al.\\citep{pal_generating_2005} discussed a method to construct random attractors to examine GNR. To our knowledge, the combination of random assignment of operators and the frequentist approach to operators have not been reported. We consider this method can be applied not only to genes, but also to medical tests because the BMCM can expand the 2 by 2 contingency table. This expansion enables us to discuss the odds ratio in multivariate cases. A logistic regression model is also used to estimate the odds ratio in multivariate cases. Using a logistic regression model, we can calculate additive interactions of variables \\citep{hosmer_applied_2013}. However, when we assume explanatory variables are independent of each other, we cannot consider non-additive interactions. Pepe et al.\\citep{pepe_limitations_2004} pointed out that there was a pitfall in using the logistic regression model for medical markers. They argued that strong associations are required for meaningful classification accuracy in using the logistic regression model. The BMCM has a disadvantage in weighting variables, similar to the logistic regression model, whereas it has an advantage in modeling interactions. Thus, the BMCM can be an option to evaluate medical markers.\nThis study has some limitations. First, this method can be applied only to binomial data. Second, an interpretation of results can be complex (Appendix A), as the number and order of variables and positions of parentheses increase exponentially. Moreover, variables can be incommutable.\nThird, when a contingency table is written, we can use different functions. For example, we can choose a function to maximize sensitivity and choose another function to maximize specificity. There can be many cross tables and there may be a person who does not belong to any faithful categories. In this case, we should index the number of stateless persons. To clarify properties of this method, further study should be done.\n\n\\section{Conclusion}\n\nWe introduced a method, BMCM, to determine operators between binomial variables using a frequentist approach. This method can expand a 2 by 2 contingency table.\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Introduction}\n \\label{sec:intro}\nIn the framework of ultracold atom physics, the experimental tunability of\ncontrol parameters pertaining each model Hamiltonian has provided a powerful\ntool to investigate situations of fundamental physical interest \n\\cite{A:Jaksch}. \n\nOne of the most intriguing features about ultracold atoms is the possibility\nto engineer a defect-free periodic potential, as opposed, for instance, to the\ntypical framework of solid-state physics. However, on the one hand the\ninterplay between disorder and interactions in Bosonic systems has attracted\nmuch theoretical attention since the seminal work by Fisher \\cite{A:Fisher},\nand on the other several techniques such as laser speckle field \\cite{A:Lye},\nthe superposition of different optical lattices with incommensurate lattice\nconstants \\cite{A:Roth03,A:Damski2003,A:Fallani}, have proven the\nexperimental relevance of disordered systems of ultracold atoms.\n\n\nIn the present paper we will deal with the effect of disorder on the\nzero-temperature phase diagram of bosonic atoms loaded onto a 1D lattice whose\nproperties can be described in terms of Bose-Hubbard Hamiltonian. In\nparticular, we will focus on the case where the disorder affects the hopping\nterm\n\\begin{eqnarray}\n \\label{eq:BHdis}\n H =&& \\sum_{m=1}^M \\frac{U}{2} a_m^\\dag a_m^\\dag a_m a_m \n -\\mu\\, a_m^\\dag a_m \\nonumber \\\\ \n && -\\sum_{m,m'} J_{m,m'} a_m^\\dag a_{m'} + h.c. \n\\end{eqnarray}\nwhere $a_m^\\dag$ ($a_m$) represents the creation (destruction) operator on\nsite $m$. The Hamiltonian parameters $U$, $J_{m,m'}$ represent the two-body\ninteraction and the (random) hopping amplitude between neighboring sites\nrespectively.\n\nIn the recent past, many authors have approached the analysis of the disordered\nBH model with various techniques such as field-theoretic approaches\n\\cite{A:Fisher,A:Wallin,A:Svistunov1996,A:Pazmandi},\ndecoupling (or Gunzwiller) mean-field approximations\n\\cite{A:Damski2003,A:Sheshadri1993,A:Krauth1992,A:Sheshadri1995,A:Krutitsky},\nquantum Monte-Carlo simulations\n\\cite{A:Scalettar1991,A:Lee2004},\namong many others, e.g.\n\\cite{A:Singh,A:Freericks1996,A:Pai,A:Rapsch,A:Pugatch}.\n\nFollowing \\cite{A:Buonsante06CM}, we employ a site-decoupling mean-field\napproximation (SDMFA), which allows to capture all of the essential features\nof the phase diagram of the model \\eqref{eq:BHdis}. The phases of the\nzero-temperature phase diagram are determined through the calculation of two\ndifferent observables: the condensate fraction, defined as the largest\neigenvalue of the one-body density matrix and the superfluid fraction, defined\nas the system response to the coupling to an external field \\cite{A:Roth03,A:Penrose}.\n\nAt zero temperature, as already pointed out in \\cite{A:Fisher} for the on-site\ndisordered BH model, we expect the presence of three phases: the\nMott-Insulating (MI) phase, where both condensate fraction and superfluid\nfraction are zero, the superfluid phase, where both superfluid phase and\ncondensate fraction are different from zero, and, finally, the Bose-Glass\nphase, which has zero superfluid fraction and finite condensate fraction,\nwhich represents a typical feature of disordered hopping systems.\n \n \nIn Section \\ref{sec:MF}, we introduce the site decoupling mean-field\napproximation for the case given by Hamiltonian \\eqref{eq:BHdis} and we depict\nthe zero-temperature phase diagram, discussing similarities and differences\nbetween the random-hopping and the random on-site potential case.\n \nIn Section \\ref{sec:stab}, we discuss the stability of the Mott phase through\nthe stability analysis of the recurrence map induced by the SDMFA. This\nanalysis will be performed comparing the results obtained by numerical exact\ndiagonalization and analytical results based on random-matrix theory.\n\n\\section{Site-decoupling mean-field scheme}\n\\label{sec:MF}\n\nThe SDMFA was introduced in Ref. \\cite{A:Sheshadri1993}, and relies on the\napproximation\n\\begin{equation}\n\\label{E:mfa}\na_m^\\dag a_{m'} = a_{m'}^\\dag \\alpha_m + a_m \\alpha_{m'}^* - \\alpha_m \\alpha_{m'}^*\n\\end{equation}\nwhere the $\\alpha_m$'s are mean-field variables to be determined\nself-consistently. The above posit turns the BH Hamiltonian \\eqref{eq:BHdis}\ninto a mean-field Hamiltonian that is the sum of on-site contributions.\n\\begin{eqnarray}\n{\\cal H} & = & \\sum_m {\\cal H}_m + J \\sum_{m \\, m'} \\alpha_m^* A_{m \\, m'} \\alpha_m'\n\\label{E:MFH} \\\\\n{\\cal H}_m &=& \\frac{U}{2}n_m\\left(n_m-1\\right) -\\mu\\, n_m \\nonumber\\\\\n&-& J (\\gamma_m a_m^\\dag + \\gamma_m^* a_m) \n\\label{E:sMFH}\n\\end{eqnarray}\nwhere $n_m=a_m^\\dag a_m$ is the usual bosonic number operator and the disorder\nrelated to the hopping term has been embedded into the adjacency matrix\n$A_{m,m'}$. For nearest-neighbor hopping on a 1D system the adjacency matrix\ncan be written as\n\\begin{equation}\n \\label{eq:adjMord2}\n A_{m',m}= e^{i\\,\\phi}s_{m-1}\\delta_{m',m-1} + e^{-i\\,\\phi}s_m\\delta_{m',m+1}\n\\end{equation}\nwhere $\\phi$ takes into account a possible coupling to an external field\n(\\textit{Peierls factors}), $s_m \\in \\mathbb{R}$ accounts for a possible\ninhomogeneity of the hopping amplitude. Here we consider a 1D lattice\ncomprising $N$ sites with periodic boundary conditions, i.e.\n\\begin{equation}\n \\label{eq:pbc}\n s_{N+1}=s_1, \\quad s_0=s_{N}.\n\\end{equation}\n\nThe ground state of Hamiltonian (\\ref{E:MFH}) is\nclearly a product of on-site states,\n\\begin{equation}\n\\label{E:MFgs}\n| \\Psi \\rangle = \\bigotimes_m | \\psi_m \\rangle\n\\end{equation}\nand the requirement that the relevant energy is minimized results in the\nself-consistency constraint \\cite{B:Sachdev}\n\n\\begin{equation}\n\\label{E:SCc}\n\\gamma_m = \\sum_{m'} A_{m \\, m'} \\alpha_m, \\qquad \\alpha_m = \\langle \\psi_{m}| a_{m} |\\psi_{m}\\rangle\n\\end{equation}\nThe decoupling mean-field approach has proved to provide satisfactory\nqualitative phase diagrams for homogeneous lattices\n\\cite{A:Sheshadri1993,A:vanOosten,A:Ferreira,A:Bru,A:Buonsante04} and superlattices\n\\cite{A:Buonsante04,A:Rey03}. \n\n\\subsection{Numerical Simulation}\n\\label{sec:num}\nIn the present section we report the results of the application of SDMFA to\nthe zero-temperature phase diagram calculation.\n\nFor our calculations we have considered a lattice formed by 100 sites, with\nvalues of the adjacency matrix given by\n\\begin{equation}\n \\label{eq:hop}\n A_{m,m'}= (1+\\Delta_m)\\delta_{m',m-1}+(1+\\Delta_{m+1})\\delta_{m',m+1} \n\\end{equation}\nwhere $\\Delta_m$ is an uncorrelated random variable uniformly distributed\nbetween $\\Delta_{\\max}$ and $-\\Delta_{\\max}$, with the\nconstraint $\\Delta_{\\max}<1$ in order to preserve $J_{m,m'}>0$. In Fig.\n\\ref{fig:hop}, we have represented two explicit realizations of disorder taken\ninto account for subsequent mean-field calculations of phase diagrams.\n\n\\begin{figure}\n \\label{fig:hop}\n \\epsfig{figure=LasPhys-Penna-Fig1.eps,width=0.5\\textwidth}\n\\caption{Values of $\\Delta_m$ taken into account for the determination of the\n phase diagrams. Upper plot has weaker disorder amplitude ($\\Delta_{\\max}=0.1$)\n compared to the lower one ($\\Delta_{\\max}=0.2$).}\n\\end{figure}\n\n\nThe value of $\\Delta_{\\max}$ has been kept small enough to ensure reasonable\nself-averaging for the system under investigation. Higher disorder amplitudes\nwould require larger chains in order to obtain results independent of the\nspecific disorder realization.\n\nThe different phases have been identified through the evaluation of two\nobservables, namely the \\textit{condensate fraction}, defined as the largest\neigenvalue of the one-body density matrix\n\\begin{equation}\n \\label{eq:condfrac}\n f_c=max(\\rho_{m,m'}), \\qquad \\rho_{m,m'}=\\langle a^\\dagger_m a_m' \\rangle\/N,\n\\end{equation}\nand the \\textit{superfluid fraction} defined as\n\\begin{equation}\n \\label{eq:sffrac}\n f_s=\\lim_{\\phi \\rightarrow 0} \\frac{E(\\phi)-E(0)}{J \\langle N \\rangle\\phi^2}.\n\\end{equation}\nwhere $E(\\phi$ is the ground-state energy corresponding to the presence of a\nPeierls phase $\\phi$ in the hopping term \\eqref{eq:adjMord2}. \n\nThe MI phase is characterized by $f_c=f_s=0$,\nthe superfluid phase (SF) by $f_c \\neq 0$ and $f_s \\neq 0$, while the phase where\n$f_c\\neq 0$ and $f_s = 0$ is recognized as the Bose-glass (BG) phase \\cite{A:Fisher}. \n\nIn absence of disorder, the variation of the control parameters -- chemical\npotential and hopping amplitude-- always leads to a transition from a phase\nwith both $f_s$ and $f_c$ equal to zero to a phase with $f_s$ and $f_c$\ndifferent from zero, excluding then the presence of a Bose-glass phase.\n\nOn the other hand, if on-site disorder is present, the MI phase is (possibly)\nseparated from the SF phase by a BG phase. Likewise, when disorder affects the\nhopping term alone a BG crops up, as it is shown in Fig. \\ref{fig:lobe}.\nHowever the distribution of the BG phase in the parameter space is\nqualitatively different. For example, with on-site disorder MI and SF phase\nare separated by a BG phase as $J$ goes to zero, while with disorder on the\nhopping term the BG phase tends to disappear for small $J$\n\\cite{A:Buonsante06CM}. This region of the phase diagram seems to be a good\nstarting point for future investigations by means of a cluster MF approach\n\\cite{A:loophole,A:LoopLP}, because it would reveal possible MI phases\nthat are not detectable through the single-site mean-filed technique\nimplemented in the present paper.\n\nIn Fig. \\ref{fig:lobe} we have represented the first lobe of the zero\ntemperature phase diagram as obtained by SDMFA for $\\Delta_{\\max}=0.1$ and\n$\\Delta_{\\max}=0.2$. \n\n\\begin{figure}[t!]\n \\label{fig:lobe}\n \\epsfig{figure=LasPhys-Penna-Fig2_1.eps,width=0.5\\textwidth}\n \\epsfig{figure=LasPhys-Penna-Fig2_2.eps,width=0.5\\textwidth}\n\\caption{Plot of the zero-temperature phase diagram for\n $\\Delta_{\\max}=0.1$(upper panel) and $\\Delta_{\\max}=0.2$ (lower panel). In\n black we have plotted the border between the MI and the BG phase, while the\n colored background corresponds to the superfluid fraction (grey=$0$). It is\n possible to notice how the BG phase covers an increasing area as\n $\\Delta_{\\max}$ is increased.}\n\\end{figure}\n\n\\section{Stability of the Mott phase}\n\\label{sec:stab}\nIn the present Section we introduce a procedure to determine the border of the\nMI phase which is based on the stability of the self-consistency map induced\nby the mean-field procedure.\n\nAs a first consideration, it is possible to state that the condition $\\gamma_k\n= 0$ for every $k$ corresponds to the gapped insulating phase of the\nmean-field Hamiltonian \\ref{E:MFH}. Indeed, in this case the local\nground-states \\eqref{E:MFgs} are eigenvectors of the local number operator\n$a^\\dag a$\n\\begin{equation}\n\\label{E:gsAc2}\n|\\psi \\rangle = |n\\rangle = \\frac{(a^\\dag)^n}{n!}|0\\rangle,\\quad n = \\lceil \\frac{\\mu}{U} \\rceil,\n\\end{equation}\nwhere $\\lceil x \\rceil$ denotes the smallest integer larger than $x$, and\nhence the mean-field ground-state (\\ref{E:MFgs}) is a pure Fock state. As for\nthe ordered Bose-Hubbard Hamiltonian, the relevant on-site energy is\n\\begin{equation}\n\\label{E:lgsE}\n\\epsilon_n = \\frac{U}{2} n(n-1) + \\mu n \n\\end{equation}\nHence $\\langle \\psi|a|\\psi \\rangle = 0$ at every site, and the\nself-consistency constraint \\eqref{E:SCc} is satisfied. In other terms\n$\\gamma_k = 0$ is a fixed point of the map defined by Eq. \\eqref{E:SCc}. The\ngapped insulating phase is stable as long as this fixed point is stable. This\nis true as long as the magnitude of the maximal eigenvalue of the matrix\n$\\Lambda$ appearing in the linearized version of Eq.\\eqref{E:SCc} is smaller\nthan 1, see \\eqref{eq:linSelfC}. In order to determine $\\Lambda$, we assume\n$|\\gamma_k| \\ll 1$ and\nconsider the (mean-field) kinetic term in Hamiltonian \\eqref{E:sMFH} as\nperturbative. \nIf first order perturbation theory is performed one gets\n\\begin{equation}\n\\label{eq:linSelfC}\n\\langle a_m \\rangle = \\frac{J}{U} F\\left(\\frac{\\mu}{U}\\right) \\sum_{m'} A_{m\\,m'} \\langle a_{m'} \\rangle\n\\end{equation}\nwhere\n\\begin{equation}\nF\\left(x\\right) = \\frac{x+1}{(\\lceil x\\rceil-x)(x-\\lfloor x\\rfloor)}\n\\end{equation}\nHence the linearized version of the self-consistency map Eq. \\eqref{E:SCc} can be written as\n\\begin{equation}\n\\langle a_m \\rangle = \\frac{J}{U} \\sum_{m'} \\Lambda_{m\\,m'} \\langle a_{m'} \\rangle \n\\end{equation}\nwhere the matrix $\\Lambda$ is proportional to the adjacency matrix $A$:\n\\begin{equation}\n\\Lambda = F\\left(\\frac{\\mu}{U}\\right) A.\n\\end{equation}\nRecalling the criteria for the stability of linear maps, the fixed point\n$\\langle a_m \\rangle = 0$ (equivalent to $\\gamma_m = 0$) is stable whenever\n\\begin{equation}\n\\label{eq:StabCond}\n\\frac{J}{U}\\leq \\frac{1}{|\\tilde{\\lambda}_{\\max}|}\n\\end{equation}\nwhere $\\tilde{\\lambda}_{\\max}$ the eigenvalue of $\\Lambda$ with the largest\nmagnitude. \n\nThe problem of determining the maximal eigenvalue of the matrix $\\Lambda$ can\nbe reduced to the calculation of the maximal eigenvalue of the ( tridiagonal )\nadjacency matrix $A$. Note that $A$ is basically the one-particle Hamiltonian\nfor a non-interacting off-diagonal Anderson model (random hopping model). If\n$\\lambda_{\\max}$ is the maximal eigenvalue of the adjacency matrix, Eq.\n\\eqref{eq:StabCond} can be recast as\n\n\\begin{equation}\n \\label{eq:StabCond2}\n \\frac{J}{U}\\leq \\frac{1}{|\\lambda_{\\max}|F\\left(\\frac{\\mu}{U}\\right)}.\n\\end{equation}\n\n\nWe have dealt with the calculation of the eigenvalues of $A$ both numerically\nand analytically. The analytical approach consists in the determination of\nthe spectral density of the matrix, given the probability distribution of the\nelements of the matrix following the approach outlined by Dyson for and\nharmonic-oscillator chain \\cite{A:Dyson}, while the former simply consists in the direct\nnumerical evaluation of the matrix eigenvalues.\n \n\n\\subsection{Numerical analysis}\n\\label{sec:NumAn}\nIn Fig. \\ref{fig:lmax} we report the maximum eigenvalue $\\lambda_{\\max}$ for\nthe random adjacency matrix $A$ as a function of the strength of disorder for\nboth a single realization and a disorder average over 1000 samples. The two\npanels refer to different lattice sizes. In every case, it is evident that the\nmaximal eigenvalue $\\lambda_{\\max}$ grows with increasing disorder strength,\n$\\Delta_{\\max}$ .\n\n\\begin{figure}[t!]\n \\centering\n \\epsfig{figure=LasPhys-Penna-Fig3.eps,width=0.5\\textwidth}\n \\caption{$\\Delta_{\\max}$-dependence of the largest eigenvalue of the\n adjacency matrix $A$, for 200 sites (upper panel) and 100 sites (lower\n panel). By comparison between the single realization plots (full lines)\n and the averaged ones (dashed lines),it is possible to see how,\n increasing the lattice size, the effect of the specific randomness\n realization is decreased}\n \\label{fig:lmax}\n\\end{figure}\n\nThe stability condition given by Eq. \\eqref{eq:StabCond2}, along with the\nabove considerations about the $\\lambda_{\\max}$-dependence from\n$\\Delta_{\\max}$ are in agreement with the considerations of Section\n\\ref{sec:num} , as far as the MI phase is concerned. In particular\n$\\lambda_{\\max}$ can be thought of as a shrinking factor for the MI lobe when\ncompared to a homogeneous situation.\n\\begin{figure}[t!]\n \\centering\n \\epsfig{figure=LasPhys-Penna-Fig4.eps,width=0.5\\textwidth}\n \\caption{Boundary of the MI region for different values of $\\Delta_{\\max}$.}\n \\label{fig:lobStab}\n\\end{figure}\n\n\\subsection{Analytical solution}\n\\label{sec:An}\n\nIn this section we would like to describe an analytical method to obtain the\nlargest eigenvalue of a (possibly infinite) random adjacency matrix whose\nentries are given by Eq. \\eqref{eq:hop}. For some particular disorder\ndistributions it is possible to carry through the analytical calculation and,\nas a consequence, solve Eq. \\eqref{eq:StabCond2} without finite-size\neffects.\n\nIt is worth mentioning that, in principle, the information gained through this\napproach is richer. In fact for some specific realizations of disorder it is\npossible to obtain the full integrated density of states $M(z)$ and not simply\nthe largest eigenvalue of the adjacency matrix.\n \nThe solution of the problem can be obtained following the approach proposed by\nDyson for the solution of a linear chain of harmonic oscillators.\n\nHere we will sketch Dyson's approach, outlining the connection with our\nproblem, providing the portion of $M(z)$ needed to obtain the largest\neigenvalue in view of Eq.\n\\eqref{eq:StabCond2}.\n\nIn \\cite{A:Dyson}, the problem of a linear harmonic chain with springs of\nrandom elastic is reformulated as a tridiagonal matrix diagonalization\nproblem. The matrix $\\Lambda$ has the following form\n\\begin{equation}\n \\label{eq:SpM}\n \\Lambda_{J+1,J}=-\\Lambda_{J,J+1}=i\\lambda_J^{1\/2}\n\\end{equation}\nwhich is related to the matrix $A$ defined in Eq. \\eqref{eq:adjMord2} by a\nunitary transformation $U(\\theta)$\n\\begin{equation}\n \\label{eq:U}\n U_{J,K}(\\theta)=\\delta_{J,K}\\exp(i \\theta J)\n\\end{equation}\nwith\n\\begin{equation}\n \\label{eq:LUAU}\n \\Lambda=U(\\theta)\\,A\\,U(\\theta)\n\\end{equation}\nand \n$$\n\\theta=\\frac{\\pi}{2}-\\phi \n$$\nand setting $s_J=\\lambda_J^{1\/2}$ in Eq. \\eqref{eq:adjMord2}. The\nunitarity of $U(\\theta)$ allows to state that the eigenvalues of $\\Lambda$ are\nequal to the eigenvalues of $A$, hence the procedure followed in\n\\cite{A:Dyson} can be directly mapped onto our problem.\n\nThe core of this approach resides in the definition of the characteristic\nfunction of the chain\n\\begin{equation}\n \\label{eq:Om}\n \\Omega(x)=\\lim_{N\\rightarrow \\infty}\\frac{\\sum_j \\log(1+x\\omega_j^2)}{N}\n\\end{equation}\nwhere $N$ is the size of the matrix, and $\\omega_j$ the desired eigenvalues of\nthe matrix under consideration.\n\nThe density of states $D(z)$ and the integrated density of states \n$$\nM(z)=\\int_0^z D(z') dz'\n$$\ncan be obtained from the characteristic function\nthrough the following relation\n\\begin{equation}\n \\label{eq:D}\n D(z)=-\\frac{1}{z^2 \\pi}{\\rm Im}\\left[\\lim_{\\epsilon \\rightarrow 0 } \\Omega'(-\\frac{1}{z}+i\\epsilon)\\right]\n\\end{equation}\nhaving defined \n$$\n\\Omega'(x)=\\frac{d \\Omega}{d x}.\n$$\n\nThe determination of $\\Omega(x)$ is obtained through a power series\nexpansion \n\\begin{equation}\n \\label{eq:OmPow}\n \\Omega(x)=\\lim_{N \\rightarrow \\infty} \\frac{1}{N} \\sum_{n=1}^\\infty (-1)^{n-1} Tr(\\Lambda^{2n}). \n\\end{equation}\nThe determination of $Tr(\\Lambda^{2n})$ leads to the following relation\n\\begin{equation}\n \\label{eq:OmXi}\n \\Omega(x)=\\lim_{N \\rightarrow \\infty} \\frac{1}{N} \\sum_{a=1}^{N}\n \\log\\left[1+\\xi(a) \\right] \n\\end{equation}\nhaving defined $\\xi(a)$ through the continued fraction\n\\begin{equation}\n \\label{eq:Xi}\n \\xi(a)=x\\lambda_a\/\\left(1+x\\lambda_{a+1}\/\\left(1+x\\lambda_{a+2}\/\\left(\\dots\\,\n \\right.\\right.\\right..\n\\end{equation}\nIf, as it may be safely assumed in our case, the various values of $\\lambda_a$\nare independent random variables with probability distribution $G(\\lambda)$,\nthe variable \n$$\n\\xi(a)=\\frac{x\\lambda_a}{1+\\xi(a+1)}\n$$ \nwill have probability distribution $F(\\xi)$ satisfying the following integral\nequation\n\\begin{equation}\n \\label{eq:Fint}\n F(\\xi)=\\int_0^\\infty F(\\xi') G\\left[\\xi(1+\\xi'\/x)\\right]\\frac{1+\\xi'}{x} d\\xi'.\n\\end{equation}\nWith the normalization condition\n\\begin{equation}\n \\label{eq:NormF}\n \\int_0^\\infty F(\\xi)d\\xi=1\n\\end{equation}\nwe obtain \n\\begin{equation}\n \\label{eq:OmUnc}\n \\Omega(x)=2\\int_0^\\infty F(\\xi)\\log(1+\\xi)d\\xi. \n\\end{equation}\n\nIf we assume a Poissonian form for $G(\\lambda)$ \n\\begin{equation}\n \\label{eq:Gpois}\n G_n(\\lambda)=\\frac{n^n}{(n-1)!}\\lambda^{n-1}\\exp(-n\\lambda),\n\\end{equation}\nEq. \\eqref{eq:Fint} has an analytical solution. Hence it is possible to obtain \nthe integrated density of states in closed form in terms of integral functions. \n\nThe integrated density of states for\n$A$ can be simply obtained back by posing\n\\begin{equation}\n \\label{eq:Ma}\n M^A(z)=M(z^2)\n\\end{equation}\nsince in \\cite{A:Dyson} $M(z)$ is defined as the proportion of eigenvalues for\nwhich $\\omega_j^22.\n \\end{array}\n\\end{equation}\nwhich, as expected, coincides with the well known result for a 1D homogeneous system.\n \nOn the other hand, if we are interested in the determination of the maximal\neigenvalue of $A$ in presence of (weak) disorder whose distribution can be\nrelated to that expressed by Eq. \\eqref{eq:Gpois}, we can consider a large-$n$\nexpansion of $M^A_n(z)$ for $z>2$.\nThe expression of $M^A_n$ in this case is given by\n\\begin{equation}\n \\label{eq:LargeM}\n M^A_n(z)\\simeq 1-\\frac{\\alpha}{\\pi}\\exp\\left[-\\alpha-2n\\left(\\sinh\n \\alpha-\\alpha \\right)\\right] \n\\end{equation}\nwith $\\alpha=\\textrm{arccosh}\\left(1\/2z^2-1\\right)$.\n\\begin{figure}[t!]\n \\centering\n \\epsfig{figure=LasPhys-Penna-Fig5.eps,width=0.5\\textwidth}\n \\caption{Comparison between three analytical solutions for $M(Z)=1$, it is\n possible to see how, for decreasing disorder ($n$ increasing) the solution\n approaches the solution of the problem without randomness. In the $x$-axis\n the variable $z$ has been normalized with respect to the maximum\n eigenvalue $\\lambda_{max}$ }\n \\label{fig:intD2}\n\\end{figure}\nFig. \\ref{fig:intD2} shows the behavior of the integrated density of states\nin the vicinity of the band edge at $\\lambda_{\\max}$, for three different\nvalues of the disorder strength. In Ref. \\cite{A:Buonsante06CM} the behavior\nof the corresponding density of states is related to the presence of a BG\nphase outside the MI region. In that case a possible direct transition from\nthe MI to the SF phase is possible for small disorders and specific values of\nthe chemical potential, and signaled by a singularity at the band edge in the\ndensity of states, similar to the Van Hove singularity characterizing the\nhomogeneous case, Eq. \\eqref{eq:Minf}. Conversely, in the present case, where\ndensity of states depends on the chemical potential through an overall\nmultiplicative factor, an infinitesimal disorder is sufficient to smear the\ndiscontinuity in the density of states. Hence an intermediate BG phase is\nexpected for every value of the chemical potential, which is in agreement with\nthe previously noted shape of the BG phase in Figs. \\ref{fig:lobe}and \\ref{fig:lobStab}.\n\n\\begin{figure}[t!]\n \\centering\n \\epsfig{figure=LasPhys-Penna-Fig6.eps,width=0.5\\textwidth}\n \\caption{Plot of the ``shrinking factor'' for the MI lobe boundary as a\n function of the disorder intensity ($1\/n \\rightarrow 0$ represents the case\n without disorder).}\n \\label{fig:shFact}\n\\end{figure}\n\n\n\\section{Conclusions}\n\\label{sec:concl}\nIn this paper we have considered the effect of a random hopping term on the\nzero-temperature phase diagram of the Bose-Hubbard Hamiltonian. Analogously to\nwhat happens when a random on-site term is considered, we have observed the\nemergence of a Bose-glass phase. The analysis has been performed within a\nmean-field approach both numerically and analytically. The boundaries of the\nMott lobes and the presence of a surrounding BG phase has been related to the\nspectral feature of an off-diagonal Anderson model. For the future, we plan to\nextend our research towards the finite-temperature case and to higher\ndimensions.\n\n{\\bf Acknowledgments.} One of the authors (PB) acknowledges a grant form\n\\textit{Lagrange project}-CRT Foundation and hospitality of the Ultra Cold\nAtoms group at the University of Otago. The work of F.M. has been entirely\nsupported by the MURST project Cooperative Phenomena in Coherent System of\nCondensed Matter and their Realization in Atomic Chip Devices.\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Introduction}\n\nOne of the most common tasks in combinatorics is to find explicitly the size \nof a certain finite set, depending on an integer parameter $n$ and defined in\nan intricate way. Then the next question usually asks how the sequence of\nnumbers describing this size behaves for large values of $n$. Of particular\ninterest is logarithmic behavior of the sequence (i.e. its log-convexity\nor log-concavity), since it is often instrumental in obtaining its growth\nrate and asymptotic behavior. Also, log-behavior may qualify (or disqualify)\na sequence as a candidate for use in certain models. A good example is the\nrecent use of log-convex sequences in quantum physics for constructing\ngeneralized coherent states associated with models having discrete non-linear\nspectra (\\cite{penson}).\n\nThe literature on log-behavior of combinatorial sequences is vast; we refer\nthe reader to the book \\cite{karlin}, and also to \\cite{brenti}, \\cite{stanley89}\nand \\cite{stanley00}. \n\nIn this article we quantitatively refine the concept of log-convexity by \nintroducing and considering the class of log-balanced\ncombinatorial sequences and showing that the terms of such sequences satisfy\ncertain double inequalities. We further proceed by deriving sufficient \nconditions for a (combinatorial) sequence given by a two-term linear \nhomogeneous recurrence to be log-convex and log-balanced. It is also indicated \nhow to extend this approach to longer recurrences and how to treat the case of\nnonhomogeneous recurrences. Finally, we demonstrate that the class of \nlog-balanced sequences is rich enough to include many cases of special \ncombinatorial interest. As a consequence, we obtain new pairs of inequalities \nfor many classical sequences.\n\n\\section{Log-balanced sequences}\n\nA sequence $\\left ( a_n \\right )_{n \\geq 0}$ of positive real numbers is\n{\\bf log-convex} if $a_n^2 \\leq a_{n-1} a_{n+1}$ for all $n \\geq 1$. If the\nopposite inequality, $a_n^2 \\geq a_{n-1} a_{n+1}$ is valid for all $n \\geq 1$,\nwe say that the sequence $\\left ( a_n \\right )_{n \\geq 0}$ is {\\bf log-concave}.\nIn case of equality, $a_n^2 = a_{n-1} a_{n+1}$, $n \\geq 1$, we call the\nsequence $\\left ( a_n \\right )_{n \\geq 0}$ geometric or {\\bf log-straight}.\nAnother type of logarithmic behavior is that of the Fibonacci sequence, where\ndirection of the inequality depends on the parity of $n$. We call such\nsequences {\\bf log-Fibonacci}.\n\nAn alternative way of characterizing the log-behavior of a sequence is via the\nsequence of quotients of its successive terms. We call the sequence\n$\\left ( x_n \\right )_{n \\geq 1}$, $x_n = \\frac{a_n}{a_{n-1}}$ the \n{\\bf quotient sequence} of the sequence $\\left ( a_n \\right )_{n \\geq 0}$.\nObviously, the sequence $\\left ( a_n \\right )_{n \\geq 0}$ is\nlog-convex if and only if its quotient sequence is non-decreasing. Similarly,\n$\\left ( a_n \\right )_{n \\geq 0}$ is log-concave if and only if its quotient sequence is non-increasing,\nand log-Fibonacci if and only if no three successive elements of the quotient\nsequence form a monotone subsequence.\n\nIn what follows, we consider log-convex sequences whose quotient sequence\ndoes not grow too fast. We shall also assume that $a_0 = 1$, unless explicitly\nstated otherwise. This restriction is not too severe, since in many combinatorially\ninteresting cases we put $a_0 = 1$ by convention.\n\nA sequence $\\left ( a_n \\right )_{n \\geq 0}$ of positive real numbers is\n{\\bf log-balanced} if $\\left ( a_n \\right )_{n \\geq 0}$ is log-convex and the\nsequence $\\left ( \\frac{a_n}{n!} \\right )_{n \\geq 0}$ is log-concave. In terms\nof quotient sequences, this means that $x_n \\leq x_{n+1} \\leq \\frac{n+1}{n} x_n$,\nfor all $n \\geq 1$.\n\nThe motivation for considering such sequences comes from the recent article\n\\cite{asai}, where it was shown that the sequences of Bell numbers of any\norder are of\nthis type. Since this property makes them suitable for providing important\nexamples in white noise theory (\\cite{kuo}), it is of interest to see \nwhether there are some other such sequences and to characterize them.\n\nWe start by stating in terms of log-balanced sequences the following \nobservation, made in (\\cite{asai}). The proof is reproduced here \nfor the reader's convenience.\n\n{\\bf Proposition 1}\\\\ Let $\\left ( a_n \\right )_{n \\geq 0}$ be a log-balanced\nsequence. Then \\\\\n(a) $a_n^2 \\leq a_{n-1} a_{n+1} \\leq \\left ( 1 + \\frac{1}{n} \\right ) a_n^2,\n\\quad n \\geq 1;$\\\\\n(b) $a_n a_m \\leq a_{n+m} \\leq \\binom{n+m}{n} a_n a_m, \\quad n, m \\geq 0.$\n\n{\\bf Proof}\\\\ The double inequality (a) is just another way of stating the\nfact that the sequence $\\left ( a_n \\right )_{n \\geq 0}$ is log-balanced.\n\nThe left inequality of (b) follows easily (by induction) from the log-convexity of \n$\\left ( a_n \\right )_{n \\geq 0}$. To prove the right inequality, start from\n$x_{n}\\geq \\frac{n}{n+1}x_{n+1}.$\nBy using this inequality repeatedly, we get\n$$\\frac{a_{1}}{a_{0}}\\geq \\frac{1}{2}\\frac{a_{2}}{a_{1}}\\geq \\frac{1}{3}\\frac{%\na_{3}}{a_{2}}\\geq ...\\geq \\frac{1}{m+n}\\frac{a_{m+n}}{a_{m+n-1}},$$ for all $%\nn\\geq 0$, $m\\geq 1.$\n\nHence, for any $0\\leq j\\leq m-1,$ we have $$\\frac{a_{j+1}}{a_{j}}\\geq \\frac{%\nj+1}{m+n}\\frac{a_{m+n}}{a_{m+n-1}}.$$\nFrom this we get\n$$\\frac{a_{1}}{a_{0}}\\frac{a_{2}}{a_{1}}\\frac{a_{3}}{a_{2}}...\\frac{a_{m}}{%\na_{m-1}}\\geq \\left(\\frac{1}{n+1}\\frac{a_{n+1}}{a_{n}}\\right)\\left(\\frac{2}{n+2}\\frac{a_{n+2}%\n}{a_{n+1}}\\right)...\\left(\\frac{m}{m+n}\\frac{a_{m+n}}{a_{n}}\\right).$$\nAfter the cancellations we get $$\\frac{a_{m}}{a_{0}}\\geq \\frac{n!m!}{(m+n)!}%\n\\frac{a_{m+n}}{a_{n}},$$and, taking into account the fact that $a_{0}=1,$ we\nfinally get\n$$a_{m+n}\\leq \\binom{m+n}{n}a_{n}a_{m}.$$The case $m=0$ is trivially valid\nfor all $n\\geq 0.$ \\qed\n\n\\section{Sufficient conditions}\n\nFor most sequences of combinatorial interest there are no explicit, closed\nform expressions for their elements. On the other hand, one can often find \nrecurrences and\/or generating functions for them. So, direct ways of \nestablishing the log-behavior\nof a given sequence (i.e. of proving inequalities of the type (a) from\nProposition 1) are only rarely at our disposal. Combinatorial proofs, which are \nthe most desirable, often turn out to be rather involved and\/or tricky. (A nice\nsurvey of inductive and injective proofs of log-concavity is given in\n\\cite{sagan}.) Hence, it makes sense to seek analytical methods sufficiently \nrobust,\neasy to apply and that will work for a reasonably broad class of sequences.\nHere we present one such method that works almost automatically for sequences\ngiven by recurrence relations. We start by explaining the method for the case\nof linear homogeneous recurrences of second order, and later we indicate how \nto modify this so that it can be applied also on longer and\/or nonhomogeneous\nrecurrences.\n\nLet $\\left ( a_n \\right )_{n \\geq 0}$ be a sequence of positive\nreal numbers, given by the two-term recurrence\n\\begin{equation}\na_n = R(n) a_{n-1} + S(n) a_{n-2}, \\quad n \\geq 2,\n\\end{equation}\nwith given initial conditions $a_0$, $a_1$. The quotient sequence \n$\\left ( x_n \\right )_{n \\geq 1}$ satisfies the nonlinear recurrence \n\\begin{equation}\nx_n = R(n) + \\frac{S(n)}{x_{n-1}}, \\quad n \\geq 2,\n\\end{equation}\nwith the initial condition $x_1 = a_1 \/ a_0$. We assume that the sequence\n$(x_n)$ is bounded by two known sequences, i.e. that there are sequences\n$(m_n)$ and $(M_n)$ such that $0 < m_n \\leq x_n \\leq M_n$, for all $n \\in \\N$.\nThe sequences $(m_n)$ and $(M_n)$ can usually be rather easily inferred from\nrecurrence (2), or guessed from the initial behavior of the sequence $(x_n)$,\nand then the bounding relations are verified by induction. In many cases even\nthe constant sequences $m_n = m$ and $M_n = M$ will be sufficiently good\nlower and upper bounds for $x_n$.\n\nAs the log-convexity is of considerable interest on its own, we first establish\nsufficient conditions for a sequence $(a_n)$ given by (1) to be log-convex.\nWe assume $R(n) \\geq 0$ and treat the cases $S(n) \\leq 0$ and $S(n) \\geq 0$\nseparately. The case $S(n) \\leq 0$ is simpler and we consider it first.\n\nAssume, inductively, that $x_{n_0} \\leq x_{n_0+1} \\leq \\ldots \\leq x_n$\nfor some $n_0 \\in \\N$. Expressing $x_{n+1}$ from equation (2) and taking\ninto account that $S(n+1)\/x_n \\geq S(n+1)\/x_{n-1}$, we obtain\n$$x_{n+1} = R(n+1) + \\frac{S(n+1)}{x_n} \\geq R(n+1) + \\frac{S(n+1)}{x_{n-1}}.$$\nWe want to prove that $x_{n+1} \\geq x_n$. But this will follow if we prove\nthe stronger inequality in which $x_{n+1}$ is replaced by the right hand\nside in the above inequality. Hence, consider the circumstance\n$$R(n+1) + \\frac{S(n+1)}{x_{n-1}}\\geq x_n = R(n) + \\frac{S(n)}{x_{n-1}},$$\nor, equivalently, \n$$[R(n+1)-R(n)] x_{n-1} + S(n+1)-S(n) \\geq 0.$$\nBy denoting $R(n+1)-R(n) = \\nabla R(n)$ and $S(n+1)-S(n) = \\nabla S(n)$, we get\na compact expression for the sufficient condition for the sequence $(a_n)$\nto be log-convex:\n\\begin{equation}\n\\nabla R(n) x_{n-1} + \\nabla S(n) \\geq 0, \\quad n \\geq n_0,\n\\end{equation}\nfor some $n_0 \\in \\N$. Hence, we have established the following result:\n\n{\\bf Proposition 2}\\\\ Let $(a_n)_{n \\geq 0}$ be a sequence of positive real \nnumbers given by the two-term recurrence (1), and $(x_n)_{n \\geq 1}$ its\nquotient sequence, given by (2). If there is an $n_0 \\in \\N$ such that \n$x_{n_0} \\leq x_{n_0+1}$, $R(n) \\geq 0$, $S(n) \\leq 0$, and\n$$\\nabla R(n) x_{n-1} + \\nabla S(n) \\geq 0,$$\nfor all $n \\geq n_0$, then the sequence $(a_n)_{n \\geq n_0}$ is log-convex. \\qed\n\nWhen (as is a common case) the function $R(n)$ is \nnon-decreasing, the condition (3) can be further simplified without significant\nloss of generality by assuming $\\nabla R(n) \\geq 0$ and replacing $x_{n-1}$\nby $m_{n-1}$, or even by a constant $m$:\n\\begin{equation}\n\\nabla R(n) m + \\nabla S(n) \\geq 0, \\quad n \\geq n_0.\n\\end{equation}\n\nThe case $S(n) \\geq 0$ is a bit more complicated. Again, we start from the \ninductive assumption $x_{n_0} \\leq x_{n_0+1} \\leq \\ldots \\leq x_n$ and \nwant to show that $x_{n+1} \\geq x_n$. By expressing both sides of this\ninequality via (2), we obtain\n$$R(n+1) + \\frac{S(n+1)}{x_n} \\geq R(n) + \\frac{S(n)}{x_{n-1}}.$$\nThis is equivalent to\n$$ x_n x_{n-1}\\nabla R(n) +S(n+1) x_{n-1} -S(n) x_n \\geq 0.$$\nBy adding the term $S(n) x_{n-1} - S(n) x_{n-1}$ to the left hand side of the\nabove inequality and rearranging it, we obtain\n$$\\nabla R(n) x_n x_{n-1} + \\nabla S(n) x_{n-1} \\geq S(n) (x_n - x_{n-1}).$$\nExpressing the term $x_n - x_{n-1}$ via (2) now yields\n$$x_{n-1} [\\nabla R(n) x_n + \\nabla S(n) ] \\geq S(n) \\left [ \\nabla R(n-1)\n+ \\frac{S(n)}{x_{n-1}} - \\frac{S(n-1)}{x_{n-2}} \\right ] .$$\nNow, replacing $\\frac{S(n)}{x_{n-1}}$ with $\\frac{S(n)}{x_{n-2}}$ in the right\nhand side square brackets we get a stronger inequality which can be written as\n\\begin{equation}\nx_{n-1} x_{n-2} [ \\nabla R(n) x_n + \\nabla S(n) ] \\geq S(n) [ \\nabla R(n-1) x_{n-2} + \\nabla S(n-1) ] .\n\\end{equation}\nObviously, this inequality implies $x_{n+1} \\geq x_n$, and it can serve as a\nsufficient condition of log-convexity for the sequence $(a_n)$.\n\n{\\bf Proposition 3}\\\\ Let $(a_n)_{n \\geq 0}$ be a sequence of positive real \nnumbers given by the two-term recurrence (1), and $(x_n)_{n \\geq 1}$ its\nquotient sequence, given by (2). If there is an $n_0 \\in \\N$ such that \n$x_{n_0} \\leq x_{n_0+1}$, $R(n) \\geq 0$, $S(n) \\geq 0$, and the inequality\n$$x_{n-1} x_{n-2} [ \\nabla R(n) x_n + \\nabla S(n) ] \\geq S(n) [ \\nabla R(n-1\n) x_{n-2} + \\nabla S(n-1) ]$$\nis valid for all $n \\geq n_0$, then the sequence $(a_n)_{n \\geq n_0}$\nis log-convex. \\qed\n\nAgain, in many combinatorially relevant cases where $\\nabla R(n) \\geq 0$ and\n$m \\leq x_n \\leq M$, the sufficient condition of Proposition 3 can be simplified\nto \n\\begin{equation}\nm^2 [m \\nabla R(n) + \\nabla S(n) ] \\geq S(n) [ M \\nabla R(n-1)\n+\\nabla S(n-1) ].\n\\end{equation}\n\nTypically, propositions 1 and\/or 2 are applied so that the respective \ninequalities are verified inductively for all $n \\in \\N$ greater than some\n$n_0$, and the remaining cases are then checked by hand or using some \ncomputer algebra system.\n\nNow we turn our attention to the inequality $x_{n+1} \\leq \\frac{n+1}{n} x_n$.\nAgain, we assume $R(n) \\geq 0$ and treat the cases $S(n) \\leq 0$ and $S(n)\n\\geq 0$ separately. Also, we assume that the log-convexity of the sequence\n$(a_n)$ is already established, i.e. that the sequence $(x_n)$ is increasing.\n\nWe first consider the simpler case $R(n) \\geq 0$, $S(n) \\geq 0$ and find\nthe sufficient conditions for $x_{n+1} \\leq \\frac{n+1}{n} x_n$ as follows. \nFrom the recurrence (2)\nwe have $$x_{n+1} = R(n+1) + \\frac{S(n+1)}{x_n}.$$\nSince the sequence $\\left ( x_n \\right )_{n \\geq 1}$ is non-decreasing, we have\n$$x_{n+1} \\leq R(n+1) + \\frac{S(n+1)}{x_{n-1}}.$$\nThe condition that the right-hand side does not exceed $\\frac{n+1}{n} x_n$ is\ngiven by\n$$R(n+1) + \\frac{S(n+1)}{x_{n-1}} \\leq \\frac{n+1}{n} \\left (\nR(n) + \\frac{S(n)}{x_{n-1}} \\right ),$$\nand this is equivalent to\n$$n R(n+1) x_{n-1} + n S(n+1) \\leq (n+1) R(n) x_{n-1} + (n+1) S(n).$$\nDenoting\n$$\\Delta _R (n) = \\left | {{R(n)} \\atop {R(n+1)}} \\quad {{n} \\atop {n+1}} \n\\right |, \\quad \n\\Delta _S (n) = \\left | {{S(n)} \\atop {S(n+1)}} \\quad {{n} \\atop {n+1}} \\right |,$$\nwe get our sufficient conditions in the form \n$$ \\Delta _R (n) x_{n-1} + \\Delta _S (n) \\geq 0.$$\nHence, we have established the following result:\n\n{\\bf Proposition 4}\\\\ Let $\\left ( a_n \\right )_{n \\geq 0}$ be a log-convex \nsequence of positive real numbers given by the two-term recurrence (1).\nIf there is an $n_0 \\in \\N$ such that $x_{n_0+1} \\leq \\frac{n_0+1}{n_0} \nx_{n_0}$, $R(n) \\geq 0$, $S(n) \\geq 0$, and \n$$ \\Delta _R (n) x_{n-1} + \\Delta _S (n) \\geq 0,$$ \nfor all $n \\geq n_0$, then the sequence $\\left ( a_n \\right )_{n \\geq 0}$ \nis log-balanced. \\qed\n\nThe case $S(n) \\leq 0$ is a bit more complicated. We proceed by induction on\n$n$. First we check that $x_{n_0+1} \\leq \\frac{n_0+1}{n_0} x_{n_0}$ for some\n$n_0 \\in \\N$, and suppose that $x_k \\leq \\frac{k}{k-1} x_{k-1}$ for all \n$n_0 \\leq k \\leq n$. Denoting $-S(n) = \\tilde S(n)$, we get\n$$x_{n+1} =R(n+1) - \\frac{\\tilde S(n+1)}{x_n}, \\quad \\tilde S(n+1) \\geq 0.$$\nFrom the induction hypothesis, $x_n \\leq \\frac{n}{n-1} x_{n-1}$, it follows\n$\\frac{1}{x_n} \\geq \\frac{n-1}{n}\\frac{1}{x_{n-1}}$, and hence $-\\frac{1}{x_n}\n\\leq - \\frac{n-1}{n}\\frac{1}{x_{n-1}}$. Now we have\n$$x_{n+1} =R(n+1) - \\frac{\\tilde S(n+1)}{x_n} \\leq R(n+1) - \n\\frac{n-1}{n}\\frac{\\tilde S(n+1)}{x_{n-1}}.$$\nThe right hand side does not exceed $\\frac{n+1}{n} x_n$ if\n$$ R(n+1) -\\frac{n-1}{n}\\frac{\\tilde S(n+1)}{x_{n-1}}\n\\leq \\frac{n+1}{n} \\left ( R(n) - \\frac{\\tilde S(n)}{x_{n-1}} \\right ),$$\nand this is, in turn, equivalent to\n$$\\left [ (n+1) R(n) - n R(n+1) \\right ] x_{n-1} + \n(n-1)\\tilde S(n+1) - (n+1) \\tilde S(n) \\geq 0.$$\nThe coefficient of $x_{n-1}$ is $\\Delta _R (n)$, and the rest can be written\nas\n$$\\left | {{n-1} \\atop {n+1}} \\quad {{\\tilde S(n)} \\atop {\\tilde S(n+1)}} \n\\right | = \\left | {{S(n)} \\atop {S(n+1)}}\\quad {{n-1} \\atop {n+1}} \\right |.$$\nDenoting the right hand side determinant by $\\overline {\\Delta }_S (n)$, \nwe get the desired sufficient conditions:\n$$ \\Delta _R (n) x_{n-1} + \\overline {\\Delta }_S (n) \\geq 0.$$\nWe can summarize:\n\n{\\bf Proposition 5}\\\\ Let $\\left ( a_n \\right )_{n \\geq 0}$ be a log-convex \nsequence of positive real numbers given by the two-term recurrence (1) with\n$R(n) \\geq 0$, $S(n) \\leq 0$. If there is an integer $n_0$ such that \n$x_{n_0+1} \\leq \\frac{n_0+1}{n_0} x_{n_0}$, and if the inequality \n$$ \\Delta _R (n) x_{n-1} + \\overline {\\Delta }_S (n) \\geq 0$$\nholds for all $n \\geq n_0$, then the sequence $\\left ( a_n \\right )_{n \\geq n_0}$ is log-balanced. \\qed\n\n\\section{Examples}\n\nWe now justify our introduction of log-balanced sequences by demonstrating\nthat the class is wide enough and that it includes many sequences of \ncombinatorial relevance. As a consequence, for all our examples we establish\nthe validity of inequalities from Proposition 1. The left inequalities for some\nof the considered sequences were established earlier (\\cite{aigner}, \\cite{dv}),\nbut the right inequalities are, with one exception (\\cite{doslic}), to the best \nof our knowledge, new. For more details\non all the considered sequences, we refer the reader to the book \\cite{stanleyII}\nand to the references therein.\n\nOur first example is the sequence of {\\bf Motzkin numbers} (see, e.g. Ex. 6.38\nof \\cite{stanleyII} for its combinatorial interpretations).\n\n{\\bf Corollary 1}\\\\ The sequence $M_n$ of Motzkin numbers is log-balanced.\n\n{\\bf Proof}\\\\ The log-convexity of $M_n$ was first established\nalgebraically in \\cite{aigner}, and a combinatorial proof appeared soon\nafterwards (\\cite{callan}). By our method it follows easily by starting from \nthe recurrence\n$$M_{n}= \\frac{2n+1}{n+2}M_{n-1}+\\frac{3(n-1)}{n+2}M_{n-2}, \\quad n \\geq 2$$\nwith $M_0 = M_1 = 1$. \nHere $R(n)=\\frac{2n+1}{n+2} \\geq 0$, $S(n)=\\frac{3(n-1)}{n+2} \\geq 0$. It is\neasy to prove by induction on $n$ that $2 \\leq M_n\/M_{n-1} \\leq 7\/2$ for\nall $n \\geq 2$, and the log-convexity follows by computing $\\nabla R(n)$, \n$\\nabla S(n)$, $\\nabla R(n-1)$, and $\\nabla S(n-1)$ and then \nverifying the inequality (6). From the fact that $ \\Delta _R (n) = \n\\frac{2 n^2 + 4 n +3}{(n+2)(n+3)}\\geq 0$, \n$ \\Delta _S (n) = \\frac{n^2 -n -3}{(n+2)(n+3)}\\geq 0$ and $x_{n-1} \\geq 0$ \nfor all $n \\geq 3$,\nit follows that $ \\Delta _R (n) x_{n-1} + \\Delta _S (n) \\geq 0$ for all \n$n \\geq 3$. The log-balancedness of $(M_n)$\nnow follows from Proposition 4, after direct verification of the defining\ninequality for the remaining values of $n$. \\qed\n\nOur next example is the sequence of {\\bf Fine numbers}. The reader may \nconsult the recent survey \\cite{deutsch} for more details on Fine numbers and on\ntheir combinatorial interpretations.\n\n{\\bf Corollary 2}\\\\ The sequence $B_n$ of Fine numbers is log-balanced for\n$n \\geq 2$.\n\n{\\bf Proof}\\\\ We start from the recurrence\n$$B_n = \\frac{7n-5}{2n+2} B_{n-1} + \\frac{2n-1}{n+1} B_{n-2}, \\quad n \\geq 2,$$\nwith initial conditions $B_0 = 1$ and $B_1 = 0$. The quotient sequence, \n$x_n = B_n\/B_{n-1}$, is defined for $n \\geq 3$. It is easy to show, by \ninduction on $n$, that $3 \\leq x_n \\leq 6$ for all $n \\geq 3$. In fact, \n$3 \\leq x_{n-1} \\leq 6$ implies $3 \\leq x_n \\leq 6$ via the above recurrence \nfor $n \\geq 7$, and $x_n$ is obviously between $3$ and $6$ for $n = 3,4,5,$\nand $6$. We proceed by computing $\\nabla R(n) = \\frac{6}{(n+1)(n+2)}$,\n$\\nabla S(n) = \\frac{4}{(n+1)(n+2)}$, $\\nabla R(n-1) = \\frac{6}{n(n+1)}$, and\n$\\nabla S(n-1) = \\frac{4}{n(n+1)}$. After plugging in these expressions we\nfind, condition (6) becomes $$ 10n^2 -30 n +80 \\geq 0,$$\nand this is true for all $n \\in \\N$. Hence, the sequence $(B_n)_{n \\geq 2}$\nis log-convex. The log-balancedness now follows by computing $ \\Delta _R (n)\n= \\frac{7n-10}{2(n+2)}$, $ \\Delta _S (n)= 2 \\frac{n-1}{n+2}$, and applying\nProposition 4. \\qed\n\nThe {\\bf Franel numbers} of order $r$ are defined by\n$$ F_n^{(r)} = \\sum _{k = 0}^n \\binom{n}{k} ^r.$$\n\n{\\bf Corollary 3}\\\\ The sequences of Franel numbers of order $3$ and $4$ are\nlog-balanced.\n\n{\\bf Proof}\\\\ It is known that Franel numbers of order $r$ satisfy a \nhomogeneous linear recurrence of\norder $\\lfloor \\frac{r+1}{2} \\rfloor$ with polynomial coefficients \n(\\cite{stanleyII}, p. 245-6 and p. 278). We have\n$$F^{(r)}_n = R^{(r)}(n) F^{(r)}_{n-1} + S^{(r)}(n) F^{(r)}_{n-2}, \\quad\nr = 3, 4, \\quad n \\geq 2,$$\nwith $F_0^{(3)} = F_0^{(4)} = 1$, $F_1^{(3)} = F_1^{(4)} = 2$. Here\n$$R^{(3)}(n) = \\frac{7n^2-7n+2}{n^2}, \\quad S^{(3)}(n) = \\frac{8(n-1)^2}{n^2},$$\n$$R^{(4)}(n) = 2 \\frac{6n^3-9n^2+5n-1}{n^3}, \\quad S^{(4)}(n) = \\frac{(4n-3)(4n-4)(4n-5)}{n^3}.$$\nObviously, all coefficient functions are non-negative. \nWe work out the case $r = 3$, and leave the details for $r = 4$ to the \ninterested reader. By examining first few values of $x_n$, one can note\nthat they are slowly increasing, starting from $x_2 = 5$. Indeed, the\nbounds $5 \\leq x_n \\leq 9$ are readily established by induction on $n$ for\n$n \\geq 3$. The log-convexity now follows by computing $\\nabla R(n)$,\n$\\nabla S(n)$, $\\nabla R(n-1)$, $\\nabla S(n-1)$, and verifying the inequality (6)\nwith $m = 5$, $M = 9$.\nTo prove the log-balancedness of $(F^{(r)}_n)$ we start by computing\n$$\\Delta _{R^{(3)}}(n) = \\frac{(n-1)(7n^3+7n^2-n-2)}{n^2 (n+1)^2}, \\quad\n\\Delta _{S^{(3)}}(n) = \\frac{8(n^4-2n^3-2n^2+n+1)}{n^2 (n+1)^2}.$$\nIt is easy to check that these determinants are positive for $n \\geq 3$,\nand that the conditions of Proposition 4 are valid for $n = 2$. \n\nProof of the case $r = 4$ is a bit more technical, but it flows along the\nsame lines, and does not present any conceptual difficulties. \\qed\n\nLet us now turn our attention to the recurrences with $S(n) \\leq 0$. Such\nexamples include, among others, Schr\\\"oder numbers, Delannoy numbers and, more\ngenerally, sequences of values of Legendre polynomials. We start with a \nsequence closely connected with Franel numbers of order $3$.\n\nThe {\\bf Ap\\'ery numbers}, $(A_n)_{n \\geq 0}$, given by the formula\n$$A_n = \\sum_{k = 0}^n \\binom{n}{k}^2 \\binom{n+k}{k}^2 =\n\\sum _{k = 0}^n \\frac{[(n+k)!]^2}{(k!)^4 [(n-k)!]^2},$$\narose in Ap\\'ery's proof of\nirrationality of $\\zeta (2)$ and $\\zeta (3)$. They are connected with Franel\nnumbers of order $3$ via the identity\n$$ A_n = \\sum _{k = 0}^n \\binom{n}{k} \\binom{n+k}{k} F^{(3)}_k, \\quad n \\geq 0$$\n(see \\cite{strehl} for history of this result). The first few Ap\\'ery numbers\nare $1, 5, 73, 1445, 33001, 819005, \\ldots $.\n\n{\\bf Corollary 4}\\\\ The sequence $A_n$ of Ap\\'ery numbers is log-balanced.\n\n{\\bf Proof}\\\\ We start from the recurrence\n$$A_n = \\frac{34n^3 - 51n^2 +27n -5}{n^3} A_{n-1} - \\frac{(n-1)^3}{n^3} A_{n-2},\n\\quad n \\geq 2,$$\nwith initial conditions $A_0 = 1$, $A_1 = 5$ (\\cite{beukers}). \nIt is easy to prove by induction on $n$ that $x_n \\geq\n1$, i.e. that the sequence of Ap\\'ery numbers is increasing. Hence we may take\n$m = 1$ as the lower bound for $x_n$. Now the expression $\\nabla R(n)\n+ \\nabla S(n)$ can serve as a lower bound for the expression (4), and the\nlog-convexity of Ap\\'ery numbers follows from the inequality\n$$\\nabla R(n) + \\nabla S(n) = \\frac{1}{n^3 (n+1)^3} [50 n^4 + 52 n^3 - 10 n^2\n-12 n + 4 ] \\geq 0,$$\nvalid for all $n \\geq 0$. \nFor the rest, first note that $x_3 = \n\\frac{1445}{73} \\leq \\frac{3}{2} x_2$, so we can take $n_0 = 2$. After\ncomputing $\\Delta _R (n)$ and $\\overline{\\Delta }_S (n)$, we get\n$$\\Delta _R (n) = \\frac{34n^6 - 72n^4 -28n^3 +27n^2 + 7n -5}{n^3 (n+1)^3},\n\\quad \\overline{\\Delta }_S (n) = \\frac{(n-1)(n^2-n-1)(2n^3+n^2-n-1)}{n^3 (n+1)^3}.$$\nBoth determinants are positive for $n \\geq 2$, and the claim follows from\nProposition 3. \\qed\n\n{\\bf Corollary 5}\\\\ The sequence $r_n$ of large Schr\\\"oder numbers is\nlog-balanced.\n\n{\\bf Proof}\\\\ Start from the recurrence\n$$r_n = \\frac{3(2n-1)}{n+1} r_{n-1} - \\frac{n-2}{n+1} r_{n-2}, \\quad n \\geq 2,$$\nwith initial conditions $r_0 = 1$, $r_1 = 2$ \\cite{stanleyII}. By computing \nthe first few values of $x_n = \\frac{r_n}{r_{n-1}}$, we guess the bounds\n$3 \\leq x_n \\leq 6$, and verify them by induction for all $n \\geq 2$. The\nlog-convexity of $(r_n)$ follows now by plugging the expressions $\\nabla R(n)\n= \\frac{9}{(n+1)(n+2)}$ and $\\nabla S(n)= - \\frac{3}{(n+1)(n+2)}$ in\nformula (4), together with $x_{n-1} \\geq 3$. To prove the rest, we compute\n$$\\Delta _R(n) = 6 \\frac{n-1}{n+2}, \\quad \\overline{\\Delta }_S(n) = \\frac{5-2n}{n+2}$$\nand note that $\\Delta _R(n) x_{n-1} + \\overline{\\Delta }_S(n) \\geq\n3 \\Delta _R(n) x_{n-1} + \\overline{\\Delta }_S(n) \\geq 0$ for all $n \\geq 1$.\nHence, by Proposition 5, the sequence $(r_n)$ is log-balanced. \\qed\n\nFor combinatorial interpretations of $r_n$, the reader may wish to consult Ex. 6.39 of\n\\cite{stanleyII}.\n\nOur next example is the sequence of values of Legendre polynomials in some\nfixed real $t \\geq 1$.\n\n{\\bf Corollary 6}\\\\ The sequence of values of Legendre polynomials $\\left (\nP_n(t) \\right ) _{n \\geq 0} $ is log-balanced for all real $t \\geq 1$.\n\n{\\bf Proof}\\\\ We start from Bonnet recurrence:\n$$P_n(t) = \\frac{2n-1}{n} t P_{n-1}(t) - \\frac{n-1}{n} P_{n-2}(t), \\quad n \\geq 2,$$\nwith the initial conditions $P_0(t) = 1$, $P_1(t) = t$.\nPassing to the recursion for the quotient sequence $x_n(t) = P_n(t)\/P_{n-1}(t)$\nwe can easily establish the lower bound $x_n(t) \\geq t$. By putting this \nlower bound, together with the expressions $\\nabla R(n) = \\frac{2}{n(n+1)}$\nand $\\nabla S(n)= - \\frac{1}{n(n+1)}$ in formula (4), we obtain the \nlog-convexity of the sequence $(P_n(t))_{n \\geq 0}$. Further,\nby computing $\\Delta _R(n)$ and $\\overline{\\Delta }_S(n)$ we get\n$$\\Delta _R(n) = \\frac{2n^2-1}{n(n+1)}, \\quad \\overline{\\Delta }_S(n) =\n\\frac{-2n^2+n+1}{n(n+1)}.$$\nIf we suppose that $\\Delta _R(n) x_{n-1}(t) + \\overline{\\Delta }_S(n) < 0$ for\nsome $n \\geq 2$, we get $x_{n-1}(t) < \\frac{1}{t} \\frac{2n^2-n-1}{2n^2-1} <\n\\frac{1}{t} < t$, in contradiction with $x_{n-1}(t) \\geq x_1(t) = t$.\nHence, the inequality $\\Delta _R(n) x_{n-1}(t)+ \\overline{\\Delta }_S(n) \\geq 0$\nholds for all $n \\geq 2$, and the claim again follows from Proposition 5. \\qed\n\nBy specializing the value of $t = 3$, we get the sequence of central Delannoy\nnumbers, $D_n = P_n(3)$ (\\cite{stanleyII}).\n\n{\\bf Corollary 7}\\\\ The sequence $D_n$ of central Delannoy numbers is \nlog-balanced. \\qed\n\nThe sequence $D_n$ counts the lattice paths from $(0,0)$ to $(n,n)$ using\nonly the steps $(1,0)$, $(0,1)$, and $(1,1)$. Equivalently, it counts king \npaths from the lower left to the upper right corner of an $(n+1) \\times (n+1)$ \nchess board.\n\nIn all examples considered so far, the sequence $(x_n)$ was increasing, but\nremained bounded. Our final example in this section shows that the same \nreasoning can be applied to the sequences whose quotient sequence increases\nunboundedly.\n\n{\\bf Corollary 8}\\\\ Let $(a_n)$ be the sequence counting directed column-convex\npolyominoes of height $n$. (See \\cite{barcucci} for the definition of\nthese objects.) The sequence $(a_n)$ is log-balanced.\n\n{\\bf Proof}\\\\ From the recurrence\n$$a_{n+1} = (n+1)a_n + a_1 +a_2 + \\ldots + a_n, \\quad n \\geq 3,$$\nwith initial conditions $a_1 = 1$, $a_2 = 3$, given in \\cite{barcucci}, one \ncan easily obtain the two-term recurrence\n$$a_n = (n+2) a_{n-1} - (n-1) a_{n-2}, \\quad n \\geq 3$$\nwith $a_1 = 1$, $a_2 = 3$. It can easily be shown by induction on $n$ that the\nsequence $x_n = \\frac{a_n}{a_{n-1}}$ is interlaced with the sequence $b_n =\nn+1$, i.e. that $n+1 \\leq x_n \\leq n+2$. Hence the sequence $(x_n)$ is \nincreasing, and $(a_n)$ is log-convex. Taking $R(n) = n+2$, $S(n) = -n + 1$,\nwe get $\\Delta _R(n) = 2$, $\\overline{\\Delta }_S(n) = 1-n$. Suppose that\n$\\Delta _R(n) x_{n-1}(t) + \\overline{\\Delta }_S(n) < 0$ for some $n \\geq 3$.\nIt follows that $x_{n-1} < \\frac{n-1}{2}$, contradicting the interlacing\nof $x_n$ and $b_n$. The claim now follows by checking the base of induction, i.e.\nthat $x_3 = \\frac{13}{3} \\leq \\frac{3}{2} \\cdot 3 = \\frac{3}{2} x_2$. \\qed\n\n\n\\section{Further developments}\n\nThe method exposed in Section 3 can be extended to the sequences given by a\nthree- (or more-) term recurrence in a straightforward way. As an illustration,\nwe treat here the case when all coefficient functions are positive and\nincreasing.\n\nLet $(a_n)$ be a sequence of positive real numbers given by the \nrecurrence $$a_n = R(n) a_{n-1} + S(n) a_{n-2} + T(n) a_{n-3}, \\quad n \\geq 3,$$\nwith given initial conditions $a_0$, $a_1$ and $a_2$. Then the recurrence for \nthe quotient sequence is given by \n\\begin{equation}\nx_n = R(n) + \\frac{S(n)}{x_{n-1}} + \\frac{T(n)}{x_{n-1} x_{n-2}}\n\\end{equation}\nfor $n\\geq 3$. We suppose inductively that $x_{n_0} \\leq x_{n_0+1} \\leq\n\\ldots \\leq x_n$ for some $n_0 \\in \\N$, and we want to find sufficient \nconditions for $x_{n+1} \\geq x_n$. This inequality can be stated as\n$$R(n+1) + \\frac{S(n+1)}{x_n} + \\frac{T(n)}{x_n x_{n-1}} - R(n) - \\frac{S(n)}{x_{n-1}} - \\frac{T(n)}{x_{n-1} x_{n-2}} \\geq 0,$$\nor equivalently\n$$x_n x_{n-1} x_{n-2} \\nabla R(n) + x_{n-2} [x_{n-1} S(n+1) - x_n S(n)]\n+ x_{n-2} T(n+1) -x_n T(n) \\geq 0.$$\nNow we proceed by a sequence of strengthenings of this inequality, leading to a\nsufficient condition that will be expressed in known quantities and reasonably\neasy to check. First we replace $S(n+1)$ and $T(n+1)$ by $S(n)$ and $T(n)$,\nrespectively. This yields\n$$x_n x_{n-1} x_{n-2} \\nabla R(n) + x_{n-2} S(n)(x_{n-1} - x_n) +T(n)\n(x_{n-2}-x_{n}) \\geq 0.$$\nBy adding $x_{n-1} - x_{n-1}$ to the term $x_{n-2}-x_{n}$ and grouping the\nterms accordingly, we obtain\n\\begin{equation}\nx_n x_{n-1} x_{n-2} \\nabla R(n) + [x_{n-2} S(n) +T(n)](x_{n-1} - x_n)\n+T(n)(x_{n-2}-x_{n-1}) \\geq 0.\n\\end{equation}\nLet us now look more closely at the term $x_{n-1} - x_n$. By inductive \nhypothesis, it must be non-positive, but we do not have any information about\nits magnitude. Expressing $x_{n-1}$ and $x_n$ via recurrence (7) yields\n$$x_{n-1} - x_n = - \\nabla R(n-1) + \\frac{1}{x_{n-1} x_{n-2}}\n[x_{n-1} S(n-1) - x_{n-2}S(n)] + \\frac{1}{x_{n-1} x_{n-2} x_{n-3}}\n[x_{n-1} T(n-1) -x_{n-3} T(n)].$$\nBy replacing $x_{n-1}$ in the first square brackets on the right hand side of\nthe above relation by $x_{n-2}$, and in the second square brackets \nby $x_{n-3}$, one obtains the following inequality:\n\\begin{equation}\nx_{n-1} - x_n \\geq - \\nabla R(n-1) - \\frac{1}{x_{n-1}}\\nabla S(n-1) - \\frac{1}{x_{n-1}x_{n-2}}\\nabla T(n-1).\n\\end{equation}\nSimilarly,\n\\begin{equation}\nx_{n-2} - x_{n-1} \\geq - \\nabla R(n-2) - \\frac{1}{x_{n-2}}\\nabla S(n-2) - \n\\frac{1}{x_{n-2}x_{n-3}}\\nabla T(n-2).\n\\end{equation}\nPlugging in formulae (9) and (10) in (8), we obtain the inequality\n\\begin{eqnarray*}\nx_n x_{n-1} x_{n-2} \\nabla R(n) & \\geq &[x_{n-2} S(n) +T(n)]\n\\left [ \\nabla R(n-1) + \\frac{1}{x_{n-1}}\\nabla S(n-1) + \\frac{1}{x_{n-1}x_{n-2}}\\nabla T(n-1) \\right ] \\cr\n & & + T(n) \\left [ \\nabla R(n-2) + \\frac{1}{x_{n-2}}\\nabla S(n-2) + \n\\frac{1}{x_{n-2}x_{n-3}}\\nabla T(n-2) \\right ].\n\\end{eqnarray*}\nFinally, by replacing the values of $x_n$, $x_{n-1}$, $x_{n-2}$, and $x_{n-3}$\nby their lower and upper bounds, we arrive at the following inequality:\n\\begin{eqnarray}\nm^3 \\nabla R(n) & \\geq & [M \\cdot S(n) +T(n)]\n\\left [ \\nabla R(n-1) + \\frac{1}{m}\\nabla S(n-1) + \\frac{1}{m^2}\\nabla T(n-1) \\right ] \\cr\n& &+ T(n) \\left [ \\nabla R(n-2) + \\frac{1}{m}\\nabla S(n-2) + \\frac{1}{m^2}\\nabla T(n-2) \\right ].\n\\end{eqnarray}\nObviously, inequality (11) implies inequality (8), and this one, in turn, \nimplies our initial inequality $x_{n+1} \\geq x_n$. Hence, inequality (11)\nprovides a sufficient condition of log-convexity for the sequence $(a_n)$.\n\nNow, assuming the log-convexity of $(a_n)$, by following the same\nreasoning as in the proof of Proposition 4, we obtain sufficient conditions\nof log-balancedness of $(a_n)$ in the form\n$$\\Delta _R(n) x_{n-1} x_{n-2} + \\Delta _S(n) x_{n-2} + \\Delta _T(n) \\geq 0,$$\nwhere $\\Delta _R(n)$ and $\\Delta _S(n)$ are as before, and $\\Delta _T(n)$ is\ndefined analogously.\n\nAs an illustration of this result, we prove that the sequence $(R_n)$, \ncounting the \n{\\bf Baxter permutations} of size $n$, is log-balanced. (See \\cite{stanleyII},\np. 246 and pp. 278-9, for more details on Baxter permutations.)\nThe numbers $R_n$ satisfy a third-order linear recurrence with the\ncoefficient functions given by\n$$R(n) = 2 \\frac{9n^3+3n^2-4n+4}{(n+2)(n+3)(3n-2)}, \\quad\nS(n) = \\frac{(3n-1)(n-2)(15n^2-5n-14)}{(n+1)(n+2)(n+3)(3n-2)},$$\n$$T(n) = 8 \\frac{(3n+1)(n-2)^2(n-3)}{(n+1)(n+2)(n+3)(3n-2)}$$\nWith a bit of help from a computer algebra system such as, e.g. {\\it\nMathematica}, it can be proved that $7 \\leq x_n \\leq 9$ for $n \\geq 47$. \nVerifying the inequality (11) then boils down to checking that a certain \nrational function of $n$ (with the degrees of the numerator and denominator\nequal to $12$ and $14$, respectively) is nonnegative for sufficiently large\nvalues of the argument. By substituting $n+3$ in place of $n$ it becomes\nobvious that all the coefficients become positive, and hence, the function\ncannot change the sign for $n \\geq 3$. The increasing behavior of $x_n$ for \n$n \\leq 47$ is\neasily checked by direct computation. Hence the sequence $(R_n)$ is log-convex.\nTo prove the log-balancedness, it is easy to check that all three determinants\n$$\\Delta _R(n) = \\frac{27n^5+18n^4+3n^3+76n^2+100n+16}{(n+1)(n+2)(n+3)(n+4)(3n+1)(3n-2)},$$\n$$\\Delta _S(n) = \\frac{135n^5-990n^4+87n^3+1036n^2+4n-112}{(n+1)(n+2)(n+3)(n+4)(3n+1\n)(3n-2)},$$\n$$\\Delta _T(n) = \\frac{9n^5-138n^4+349n^3-80n^2-252n-48}{(n+1)(n+2)(n+3)(n+4)(\n3n+1\n)(3n-2)}$$\nare positive for $n \\geq\n13$, and the log-balancedness of $(R_n)$ follows by directly verifying \ndefining inequalities in the remaining cases.\nAll the {\\it Mathematica} calculations necessary for verifying the above\ninequalities were performed exactly.\n\nThe scope of our approach can also be extended in another direction, namely\nto linear nonhomogeneous recurrences. Here we indicate, after the fashion \nof \\cite{dv}, how such recursions\ncan be transformed in a form suitable for application of our method. So, for \nexample, let $(a_n)$ be given by a linear nonhomogeneous recurrence of the\nfirst order\n\\begin{equation}\na_n = R(n) a_{n-1} + S(n)\n\\end{equation}\nwith the initial condition $a_0$.\nBy writing down the recurrence (12) for successive indices, multiplying and\nsubtracting as to cancel the nonhomogeneous part, one obtains the homogeneous\nsecond order linear recurrence for $a_n$:\n$$a_n=\\left [ R(n)+\\frac{S(n)}{S(n-1)} \\right ] a_{n-1}-\\frac{R(n-1)S(n)}{S(n-1)}a_{n-2}.$$\nBy denoting $R^{\\star}(n) = R(n)+\\frac{S(n)}{S(n-1)}$, \n$S^{\\star}(n) = -\\frac{R(n-1)S(n)}{S(n-1)}$, and dividing through by $a_{n-1}$,\nwe get a recurrence for $x_n$ of the type (2) and the further treatment \ndepends on the combination of signs of $R^{\\star}(n)$ and $S^{\\star}(n)$.\n\nSimilarly, for a second order linear recurrence\n$$a_n=R(n)a_{n-1}+S(n)a_{n-2}+T(n),$$\nwe obtain\n$$x_n=R(n)+\\frac{S(n)}{x_{n-1}}+\\frac{T(n)}{T(n-1)}\\left [ 1- \\frac{R(n-1)}{x_{n-1}}-\\frac{S(n-1)}{x_{n-1}x_{n-2}} \\right ]. $$\nThen we can proceed as before.\n\nFinally, a word of caution. It would be hasty to conclude, from the \ncited examples, that all combinatorially interesting sequences are log-balanced.\nFor example, the sequences $a_n = (n!)^2$, $a_n = (n-1)!$ and \n$a_n = \\sum _{k = 0}^n k!$\nare not log-balanced, since their quotient sequences grow too fast. It is also\ninteresting to note that the property of log-balancedness is not shift-invariant;\none can easily see that the sequence $(n+1)!$ is log-balanced, while $(n-1)!$\nis not.\n\nOne could, in principle, consider an alternative approach to the question of\nlog-balancedness, that is in a sense dual to ours. One could take a \nlog-concave sequence $(a_n)$ and ask for the sufficient conditions for the\nsequence $(n!a_n)$ to be log-convex. Since it appears that the log-convex\nsequences are much more common among the sequences of combinatorial interest,\nwe will not pursue this alternative approach here.\n\nThe author acknowledges the support of the Welch Foundation of Houston, Texas,\nvia grant \\# BD-0894.\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Introduction}\n\nThe effect of quenched disorder to critical phenomena in spin systems has been the subject of intense study for almost half a century. One of the central models has been the random bond Ising model (RBIM), which serves as a model for certain spin glass materials \\cite{Edwards1975, binder1986}, certain localization problems and plateau transitions in the quantum Hall effect \\cite{cho1997,merz2002} but also has been shown to be relevant for the analysis of the performance of topological quantum error correcting codes when assuming certain noise models \\cite{dennis2002topological, chubb2021statistical}. \n\nThe RBIM in flat space has been understood quite comprehensively by now: while weak disorder is irrelevant in the renormalization group sense \\cite{dotsenko1983}, increasing the disorder strength lowers the phase transition temperature until the so called ``Nishimori point'' is reached. Beyond this, the system stays disordered for all temperatures. In more than two dimensions, the system for low temperatures and large disorder is in a spin glass phase, with the Nishimori point being the tri-critical point.\n\nThe present paper is now concerned with the properties of the RBIM in curved space, which to the best of our knowledge has previously been studied only in the absence of disorder \\cite{Rietman1992, krcmar2008, mnasri2015, Jiang2018, breuckmann2020}. \nIn this limit, the model undergoes a phase transition from a paramagnetic high-temperature to a low-temperature ferromagnetic phase, just as its flat-space counterpart. \nThe transition is mean-field in nature, but surprisingly it is not located at the fixed-point of the Kramers--Wannier duality, even on self-dual hyperbolic lattices. This observation implies either the existence of a second phase transition, for which no evidence was found numerically, or a violation of self-duality of the Ising model on self-dual hyperbolic lattices.\nWe note that the existence of a second phase transition for the pure Ising model on the hyperbolic plane with free boundary condition has been proved~\\cite{Wu1996, Wu2000,Jiang2018}.\n\n\nAs we show in this work, there is an anomaly in the hyperbolic RBIM.\nIt turns out that it is not self-dual even on self-dual lattices, but, in the disorder-free limit, is related by the Kramers--Wannier duality to what we call the \\emph{dual-RBIM}.\nHence, in this paper we study both the critical properties of the random bond Ising model and its dual in hyperbolic space.\nNote that what we call the dual-RBIM it is not related to the RBIM by an exact duality in the presence of disorder.\n\nWe begin our study of both models by mapping out their phase diagrams using a combination of high-temperature series expansion techniques and Monte-Carlo simulations. We show that the RBIM realizes a paramagnetic, a ferromagnetic and a spin-glass phase with the Nishimori point as the tricritical point. All transitions (with the exception of the multicritical point) are compatible with second-order mean-field behavior. In contrast, the dual-RBIM in the disorder-free limit as well as along the Nishiori line undergoes a strongly first-order transition as evidenced through Metropolis and canonical simulations using the Wang-Landau algorithm.\nWe numerically verify the duality of the two models in the disorder-free case and show that a duality conjectured by Takeda et al. \\cite{takeda2005exact} is fulfilled only approximately.\n\nThe rest of the paper is organized as follows. In \\autoref{sec:intro_rbim} we give necessary notions and definitions; in particular the dual-RBIM is derived in \\autoref{sec:intro_duality}. In \\autoref{sec:methods} we derive the high-temperature expansion for the RBIM and give details on the Monte-Carlo simulations used. \\autoref{sec:rbim} presents the results on the phase diagram and critical properties of the random Bond Ising model and \\autoref{sec:dual-rbim} presents the same for the dual model. Finally, we discuss the relevance of our results to the decoding of hyperbolic surface codes in \\autoref{sec:qec}. We conclude in \\autoref{sec:conclusion}.\n\n\n\n\\section{The disordered Ising model and its dual in the hyperbolic plane\\label{sec:intro_rbim}}\n\n\\subsection{Hyperbolic surfaces}\\label{sec:intro_hyperbolic}\n\nThe hyperbolic plane is a 2D manifold of constant negative curvature.\nIt can be realized in terms of several models.\nHere, we will employ the \\emph{Poincar\\'e disk model}, which is defined as follows.\nConsider a disk in $\\mathbb{R}^2$ with unit radius and centered at the origin.\nLet $x$ and $y$ denote the standard coordinates of $\\mathbb{R}^2$.\nThen the hyperbolic plane is given by the set of points\n\\begin{align}\n \\mathbb{H}^2 = \\{ (x,y)\\in \\mathbb{R}^2 \\mid x^2+y^2 < 1 \\}\n\\end{align}\nwith metric given by\n\\begin{align}\\label{eq:metric}\n ds^2 = \\frac{dx^2 + dy^2}{\\left( 1-x^2-y^2 \\right)^2}\n\\end{align}\nIt is immediate from \\autoref{eq:metric} that length scales are highly distorted towards the boundary of the disk compared to the euclidean metric, see \\autoref{fig:55tessellation}.\n\nJust as regular euclidean space can be tessellated by squares, triangles or hexagons, hyperbolic space can be tessellated by regular polygons as well.\nIn fact, it turns out that hyperbolic space supports an infinite number of regular tessellations.\nWe can label regular tessellations by the \\emph{Schl\\\"afli symbol} $\\{r,s\\}$, where $r$ is the number of sides of the polygonal plaquettes and $s$ is the number of plaquettes meeting at each vertex.\nFor example, the hexagonal lattice has Schl\\\"afli symbol $\\{6,3\\}$.\nIts dual lattice can be obtained by reversing the Schl\\\"afli symbol, i.e. the triangular lattice $\\{3,6\\}$.\nThese two examples, together with the self-dual square tessellation $\\{4,4\\}$ are all the possible regular tessellations of the euclidean plane.\nThe hyperbolic plane supports any regular tessellation $\\{r,s\\}$ as long as $1\/r+1\/s < 1\/2$.\nThe $\\{5,5\\}$ tessellation of the hyperbolic plane in the Poincar\\'e disk model is shown in \\autoref{fig:55tessellation}.\n\n\\begin{figure}\n \\centering\n \\includegraphics[width=0.65\\columnwidth]{tesselation.png}\n \\caption{(a) Poincar\\'e disk model of the infinite hyperbolic plane $\\mathbb{H}^2$ with the $\\{5,5\\}$ lattice. All edges have the same length with respect to the hyperbolic metric, see \\autoref{eq:metric}.}\n \\label{fig:55tessellation}\n\\end{figure}\n\nIn order to approximate the infinite hyperbolic plane for numerical analysis, we can consider sequences of finite neighborhoods $B_R$ (discs) of increasing radii $R$.\nThis is commonly done in the context of statistical mechanics models in euclidean space for performing finite size analysis.\nThe models differ at the boundaries of the finite regions from the infinite euclidean plane.\nHowever, the effects of this deviation vanish in the thermodynamic limit as $\\operatorname{vol}(\\partial B_R) \/ \\operatorname{vol}(B_R) \\rightarrow 0$ for $R\\rightarrow \\infty$.\nThis is not the case in hyperbolic space where $\\operatorname{vol}(\\partial B_R)$ and $\\operatorname{vol}(B_R)$ have the same asymptotic scaling.\nThis means that taking finite neighborhoods with boundaries can not be used to analyze the behaviour of the infinite model.\nWe solve this problem by considering families of boundaryless, finite surfaces (supporting the same tessellation) which are indistinguishable from the infinite hyperbolic plane in local regions of increasing size at any point.\n\n\nIntroducing periodic boundary conditions is a much more subtle process in hyperbolic spaces compared to euclidean spaces. \nIn particular, closed, orientable hyperbolic manifolds have a genus that is proportional to their area.\nThis is seen most easily by considering a theorem due to Gau\\ss --Bonnet, which states that the geometry (curvature) of a 2D surface is connected to its topology.\nMore concretely, it states that for any orientable surface~$S$ of genus~$g$ it holds that\n\\begin{align}\\label{eqn:gauss_bonnet}\n 2-2g = \\frac{1}{2\\pi} \\int_S \\kappa\\, dA \n\\end{align}\nwhere on the right hand side we integrate the curvature~$\\kappa$ at every point in $S$ over the area of $S$.\nIf $S$ is euclidean, then the curvature~$\\kappa$ is equal to 0 at every point.\nFrom \\autoref{eqn:gauss_bonnet} it then immediately follows that all orientable euclidean surfaces are tori ($g=1$).\nOn the other hand, if $S$ is hyperbolic then $\\kappa = -1$ everywhere. Orientable hyperbolic surfaces hence have\n\\begin{align}\\label{eq:area_propto_genus}\n \\frac{\\operatorname{area}(S)}{2\\pi} = 2g-2\n\\end{align}\nso that larger surfaces necessarily have a higher genus.\nIn \\autoref{fig:klein_quartic} we show an example of a closed $g=3$ hyperbolic surface, called \\emph{Klein quartic}, which supports a $\\{7,3\\}$ tessellation.\n\n\\begin{figure}[h]\n \\centering\n \\includegraphics[width=0.2\\textwidth]{klein_compact.png}\n \\hfil\n \\includegraphics[width=0.2\\textwidth]{klein_fd.png}\n \\caption{A hyperbolic surface of genus 3 tessellated by the $\\{7,3\\}$-tessellation (left).\n If we cut the surface open we obtain a flat piece of hyperbolic space (right).\n The plaquettes are colored to guide they eye.}\n \\label{fig:klein_quartic}\n\\end{figure}\n\nAs it turns out, the subtlety that hyperbolic surfaces are topologically complex becomes important in the Kramers--Wannier duality.\nThis is because the Kramers--Wannier duality is sensitive to the number of closed loops (cycles) in the lattice and the higher genus of hyperbolic surfaces introduces more such loops, see discussion in \\autoref{sec:intro_duality}.\n\n\n\n\\subsection{Duality in the Hyperbolic Ising model\\label{sec:intro_duality}}\nWe consider the Ising model (for the time being \\emph{without} quenched disorder) on a lattice $\\mathcal L = (V, E, F)$. We denote by $V$ the set of vertices, by $E$ the set of edges and by $F$ the set of faces of the lattice.\nDenoting nearest neighbor bonds between two vertices $i$ and $j$ of the lattice by $\\expval{ij}$, the Hamiltonian of the Ising model is then given by\n\\begin{equation}\n\\label{eq:pure-Ising}\n H = J \\sum_{\\expval{ij}} \\sigma_i \\sigma_j,\n\\end{equation}\nwhere $\\sigma\\in\\{\\pm1\\}$ are Ising spin variables and we asume $J<0$ for ferromagnetic coupling.\n\nIn euclidean space, the Kramers--Wannier duality \\cite{kramers1941} relates the high-temperature expansion of the Ising model [\\autoref{eq:pure-Ising}] to its low-temperature expansion of the same model on the dual lattice. \n\nIn particular, Kramers and Wannier showed a exact relation the two partition functions\n\\begin{subequations}\n\\label{eq:krammers-wannier}\n\\begin{align}\n Z(T) = \\Tilde Z(T^*)\n\\end{align}\nwhere~$Z$ and~$\\Tilde Z$ are the partition functions of the Ising model on the lattice and its dual respectively and $T$ and~$T^*$ satisfy \n\\begin{equation}\n\\label{eq:krammers-wannier-T}\n \\sinh(2J\/T)\\sinh(2J\/T^*) = 1.\n\\end{equation}\n\\end{subequations}\n\nOn a self-dual lattice, $Z=\\Tilde Z$ and thus the duality [\\autoref{eq:krammers-wannier}]] constitutes an exact mapping between the behavior of the system at high and low temperature. In particular, assuming that a single phase transition occurs, this fixes the critical temperature to the fixed-point of \\autoref{eq:krammers-wannier-T}\n\\begin{equation}\n \\label{eq:krammers-wannier-Tc}\n \\sinh(2J\/T_c)\\sinh(2J\/T_c) = 1~\\Rightarrow~T_c \\approx 2.2692J.\n\\end{equation}\n\nAn open question posed by earlies studies \\cite{Rietman1992, breuckmann2020} was how \\autoref{eq:krammers-wannier-Tc} is violated in hyperbolic space. \nThat is, if the Kramers--Wannier duality [\\autoref{eq:krammers-wannier}] holds also for hyperbolic lattices, one of the following must hold: either that all self-dual hyperbolic lattices (that is tesselations of compact hyperbolic manifolds with Schl\\\"afli symbol $\\{r, s\\}$ with $r=s$) have the \\emph{same} critical temperature, given by \\autoref{eq:krammers-wannier-Tc}, or there exist \\emph{two} phase transitions, related by \\autoref{eq:krammers-wannier-T}.\nIn fact, as we will show below, the Ising model on tessellations of compact hyperbolic manifolds is not related by the Krammers--Wannier duality to the same Ising model on the dual lattice. In particular, it is not self-dual, even on self-dual lattices.\n\n\\subsubsection{Re-derivation of the Kramers--Wannier duality}\nTo understand this, let us perform a more careful re-derivation of the Kramers--Wannier duality.\nTo this end, we first consider the high-temperature expansion of the Ising model on a lattice $\\mathcal L = (V, E, F)$.\nLet $Z_1$ be the set of subsets $\\gamma \\subset E$ such that in the subgraph induced by any such $\\gamma$ every vertex has even degree.\nThe subsets $\\gamma \\in Z_1$ are called \\emph{cycles}.\nIt is well-known that the partition function can be written as a sum over the set of all cycles of the graph $Z_1$ (see e.g. \\cite[Chapter~2]{oitmaa2006}):\n\\begin{subequations}\n\\label{eq:kw-high-T}\n\\begin{align}\n\t\tZ(K) &= \\sum_{{\\sigma}\\in \\{\\pm 1\\}^N} \\prod_{(i,j)\\in E} \\exp(K \\sigma_i \\sigma_j)\\\\\n\t\t&= (\\cosh K)^{|E|} \\sum_{{\\sigma}} \\prod_{(i,j)} (1+ \\sigma_i \\sigma_j \\tanh K)\\\\\n\t\t&= 2^N (\\cosh K)^{|E|} \\sum_{\\gamma \\in Z_1} (\\tanh K)^{|\\gamma|}\n \\end{align}\n\\end{subequations}\nwhere we have defined $K=J\/T$ and by $|S|$ denotes the size of the set $S$.\n\nNote that the set $Z_1$ of cycles $\\gamma$ in \\autoref{eq:kw-high-T} includes ones that are contractible as well as ones that are non-contractible. Two examples for such cycles on a surface with genus 3, tessellated by the $\\{7, 3\\}$ tessellation (cf. \\autoref{fig:klein_quartic}), are given in \\autoref{fig:klein_quartic_dual}. On the right we show a contractible cycle on the primal lattice (solid lines) in blue. On the left we show, also in blue, a non-contractible cycle on the dual lattice (dashed lines).\n\n\\begin{figure}[h]\n \\centering\n \\includegraphics[width=0.2\\textwidth]{klein_cycle.png}\n \\hfil\n \\includegraphics[width=0.2\\textwidth]{klein_boundary.png}\n \\caption{The left shows a cycle on the dual lattice (blue) and the associate cocylce on the primal lattice (red). The right shows a boundary on the primal lattice (blue) and the associate coboundary on the dual lattice (red).}\n \\label{fig:klein_quartic_dual}\n\\end{figure}\n\nTo establish the duality, we also consider the low-temperature expansion of the Ising model, but on the dual lattice $\\mathcal L^* = (V^*,E^*,F^*) = (F,E,V)$. For regular tessellations of hyperbolic surfaces, the dual lattice is just obtained by swapping the first and second entry of its Schl\\\"afli symbol $\\{r, s\\}$. This is also indicated in \\autoref{fig:klein_quartic_dual}. The primal lattice (solid lines) is the $\\{7, 3\\}$ tessellation and its dual (dashed lines) is the $\\{3, 7\\}$ tessellation of the same surface.\n\nThe low temperature expansion follows from expressing the partition function in terms of excitations on top of the (ferromagnetic) ground state. These are given by domain walls.\nFor example, consider starting from a all-ferromagnetic state of the Ising model [\\autoref{eq:pure-Ising}] on the (dual) lattice indicated by dashed lines in \\autoref{fig:klein_quartic_dual}. The cost of flipping the spin on the central cite is given by the size of the domain wall indicated in red on the right of \\autoref{fig:klein_quartic_dual}.\nGenerally, let $B^{1*}$ be the set of all possible domain walls on the dual lattice. We can write\n\\begin{subequations}\n\\label{eq:low-T-z}\n\\begin{align}\n\\Tilde Z(K) &= \\sum_{{\\sigma}\\in \\{\\pm 1\\}^N} \\prod_{(i,j)\\in E^*} \\exp(K \\sigma_i \\sigma_j)\\\\\n&= 2 \\sum_{\\omega^* \\in B^{1*}} \\exp(K^*)^{|E^*|-2|\\omega^*|}\\\\\n&= 2 \\exp(K)^{|E^*|} \\sum_{\\omega^* \\in B^{1*}} \\exp(-2 K)^{|\\omega^*|}\n\\end{align}\n\\end{subequations}\nwhere the second equality directly follows from the definition of $B^{1*}$. In the language of homology, the set $B^{1*}$ is given exactly by the set of \\emph{coboundaries} on the \\emph{dual} lattice.\n\nThe basis of the Kramers--Wannier duality, homologically speaking, is the fact that the set of cycles $Z_1$ is in one-to-one correspondence with the set of \\emph{cocycles} $Z^{1*}$ on the dual lattice $\\mathcal L^*$. This is also indicated in \\autoref{fig:klein_quartic_dual} where we show two examples of the correspondence of cocycles (red) and cycles (blue). The left side shows a non-contractible cocycle on the primal lattice (solid, red) and the corresponding cycle on the dual (blue, dashed). The right side shows a contractible cycle (a \\emph{boundary}) on the primal lattice (red, solid) and the corresponding cocycle (a \\emph{coboundary}) on the dual lattice (red, dashed). \n\nUsing this equivalence, $Z_1 = Z^{1*}$, as well as \\autoref{eq:krammers-wannier-T}, and defining $K^*=J\/T^*$, we can then rewrite \n\\begin{align}\n Z(K) &= 2 \\exp(K^*)^{|E^*|}\\sum_{\\gamma^* \\in Z^{1*}}\\exp(-2K^*)^{|\\gamma^*|}.\n \\label{eq:dual-z}\n\\end{align}\nAbove, the right hand side is \\emph{almost} the low-temperature expansion of the Ising model on the dual lattice [\\autoref{eq:low-T-z}], at temperature $T^*$ [\\autoref{eq:krammers-wannier-T}].\nThe difference between \\autoref{eq:dual-z} and \\autoref{eq:low-T-z} is that the sum above is over all cocycles $\\gamma^* \\in Z^{1*}$ whereas the low-temperature expansion is a sum over domain walls $\\omega^* \\in B^{1*}$, that is coboundaries or ``contractible'' cocycles.\nPhysically, we can rationalize this difference by looking at the example of a non-contractible cocycle on the left of \\autoref{fig:klein_quartic_dual} (red, solid). The corresponding cocycle appears in the high-temperature expansion of the dual lattice (every vertex in it has even degree). However, there is no set of spins on vertices of the primal lattice that we could flip to get a domain of that form.\n\nHence, for Ising models on regular tessellations of closed manifolds, we have established what is the \\emph{difference} between their high-temperature expansion [\\autoref{eq:kw-high-T}] and their low-temperature expansion at the dual temperature [\\autoref{eq:dual-z}]. In the following we will show that (i) for tessellations of closed euclidean surfaces (tori), this difference vanishes in the thermodynamic limit, yielding the Kramers--Wannier duality [\\autoref{eq:krammers-wannier}], and (ii) the difference does \\emph{not} vanish for tessellations of closed hyperbolic surfaces, leading to a violation of \\autoref{eq:krammers-wannier}.\n\nNote that the contribution of any cocycle in \\autoref{eq:dual-z} has a weight $\\exp(-2K^*)^{|\\gamma^*|}$.\nFor euclidean lattices on an $L\\times L$ torus this implies that the contribution of any non-contractible cocycle is at least of order $\\order{\\exp(-2K^*)^L}$.\nFocussing on such minimal-size cocycles, of which there are $\\sim L$, the difference between \\autoref{eq:dual-z} and the low-temperature expansion of the Ising model vanishes in the thermodynamic limit\n\\begin{equation}\n Z(T) - \\Tilde Z(T^*) \\sim L \\exp(-2K^*L) \\xrightarrow[L\\to \\infty]{} 0.\n\\end{equation}\nThis then yields \\autoref{eq:krammers-wannier}.\n\nIn contrast, in hyperbolic space, the number of minimal, non-contractible cocycles goes as $\\sim N$ [see \\autoref{eq:area_propto_genus}] while their length grows only logarithmically \\cite{macaj2008injectivity,moran1997growth} and so the same difference goes as\n\\begin{equation}\n Z(T) - \\Tilde Z(T^*) \\sim N^{1-2K^*}\n\\end{equation}\nwhich does not generally vanish as $N\\to\\infty$.\n\n\\subsubsection{The dual Ising model in hyperbolic space}\n\nIn order to obtain a model that does fulfill the Kramers--Wannier duality, we have to define a model where possible domain walls on top of the ferromagnetic ground state include all non-contractible cocycles.\n\nWe achieve this by a rather simple trick. Given an Ising model [\\autoref{eq:pure-Ising}] on a tessellation of a closed hyperbolic surface~$S$ with $2g$ nonequivalent, non-contractible cocyles~$\\ell$, we introduce one additional Ising degree of freedom~$\\eta_\\ell$ per nonequivalent, non-contractible cocycle.\n\nWe then define the ``dual Ising model'' as\n\\begin{align}\\label{eq:H-dual-ising-pure}\n H &= J \\sum_{\\expval{ij}} \\left( \\prod_{\\ell\\,|\\,\\expval{ij} \\in \\ell} \\eta_\\ell \\right)\\,\\sigma_i \\sigma_j\n\\end{align}\nwhere $J<0 $ as before is chosen to be ferromagnetic and we have chosen one representative per nontrivial cocycle~$\\ell$. One example of such a representative on a hyperbolic surface with genus 3, tessellated by the $\\{7, 3\\}$-tesselation is shown on the left side of \\autoref{fig:klein_quartic_dual}, where it is highlighted in red. The effect of flipping this Ising degree of freedom $\\eta_\\ell \\to -\\eta_\\ell$ is to reverse the sign of the coupling of each edge that is part of the representative $\\ell$. One can think of each variable $\\eta_\\ell$ to encode the \\emph{boundary} condition in one possible direction which can either be periodic ($\\eta_\\ell=1$) or anti-periodic ($\\eta_\\ell=-1$). \nBecause of this, the domain walls of the model defined by \\autoref{eq:H-dual-ising-pure} include the nontrivial cocycles of the lattice and its partition is given by \\autoref{eq:dual-z}, that is the dual-RBIM for $p=0$ is indeed the Kramers--Wannier dual of the Ising model.\n\n\n\\subsection{The Random-Bond Ising model}\n\nThe random-bond Ising model (RBIM), first introduced by Edwards and Anderson \\cite{Edwards1975} to model the interaction of dilute magnetic alloys, serves as a simple model to study critical phenomena in systems with quenched disorder.\nThe Hamiltonian for the RBIM on a lattice with nearest-neighbor bonds $\\expval{ij}$ is\n\\begin{align}\n H = \\sum_{\\langle i,j \\rangle} J_{ij} \\sigma_i \\sigma_j\n \\label{eq:H-RBIM}\n\\end{align}\nwhere $\\sigma_i\\in \\lbrace \\pm 1 \\rbrace$ are Ising spin variables and $J_{ij}$ are random couplings.\nWhenever we refer to the Ising model in ``hyperbolic space'' or on ``hyperbolic lattices'' throughout this work, we refer to a model where spins are located on the vertices of regular tessellations of \\emph{compact} hyperbolic manifolds, with Schl\\\"afli symbol $\\{r,s\\}$. This emphasis is important, since considering the same model on non-compact hyperbolic manifolds with, for example, open or closed boundary conditions will generally change its properties \\cite{Wu1996, Wu2000}. \nThe couplings are distributed independently and identically. In this paper, we take their individual probability distribution to be the so called \n``$\\pm J$-distribution''\n\\begin{align}\\label{eq:disorder}\n P(J_{ij}) = p\\, \\delta(J_{ij}-1) + (1-p) \\, \\delta(J_{ij}+1)\n\\end{align}\nso that each coupling is anti-ferromagnetic $J_{ij} = +1$ with probability $p$ and ferromagnetic $J_{ij} = -1$ with probability $1-p$.\nHence, on the infinite hyperbolic plane, $p$ is equal to the fraction of anti-ferromagnetic bonds. \nThe free energy of the model, when considering quenched disorder is then given by\n\\begin{align}\n F &= \\left[\\log(Z) \\right], \\\\\n Z &= \\sum_{\\{\\sigma\\}} \\exp\\left(-\\beta \\sum_{\\langle i,j \\rangle} J_{ij} \\sigma_i \\sigma_j \\right),\n\\end{align}\nwhere brackets $[\\dots]$ denote the average over disorder configurations.\n\nFor $p=0$, the model reduces to the ferromagnetic Ising model, which we have studied for regular tessellations of compact hyperbolic manifolds in a previous paper \\cite{breuckmann2020}. This model as a function of temperature undergoes a phase transition from a high-temperature paramagnetic into a low-temperature ferromagnetic phase. Our study revealed that this transition is mean-field in nature for all investigated tessellations. In the present work, we extend our previous work to the case of finite $0 < p < 1\/2$. \n\nWe also study the \\emph{dual} Ising model \\autoref{eq:H-dual-ising-pure} in the presence of quenched disorder. In this case it becomes\n\\begin{align}\\label{eq:H-dual-ising}\n H &= \\sum_{\\expval{ij}} J_{ij} \\left( \\prod_{\\ell\\,|\\,\\expval{ij} \\in \\ell} \\eta_\\ell \\right)\\,\\sigma_i \\sigma_j.\n\\end{align}\nAs before, the $\\sigma_j\\in\\{\\pm1\\}$ are Ising variables, as are the $\\eta_\\ell \\in \\{\\pm 1\\}$. While the $\\sigma_j$ are located on the vertices of the lattice, each $\\eta_\\ell$ is associated with a nontrivial cocycle $\\eta_\\ell$ (cf. \\autoref{sec:intro_duality}). The $J_{ij}$ are random couplings drawn from the $\\pm J$ distribution defined in \\autoref{eq:disorder}.\n\nThe Kramers--Wannier duality [\\autoref{eq:krammers-wannier}], as usual, is only valid is the disorder-free case.\nHowever there is conjecture by Takeda and Nihsimori \\cite{takeda2005exact} relating the location of the Nishimori point of the RBIM with the position in the dual model\n\\begin{equation}\\label{eq:duality-conjecture}\n H(p_{\\rm N}) + H(p_{\\rm N}^*) = 1\n\\end{equation}\nwhere $H(p) = -p \\log_2(p) - (1-p)\\log_2(1-p)$ is the binary entropy.\nAs discussed in \\autoref{sec:dual-rbim} we see that the conjecture holds approximately, but not within error bars.\n\n\n\n\n\n\\subsection{Possible Phases and Order Parameters}\n\n\\begin{figure}\n \\centering\n \\includegraphics{phases_sketch.pdf}\n \\caption{Schematic phase diagram of the random bond Ising model and its dual on the hyperbolic plane as a function of temperature $T$ and the fraction of antiferromagnetic bonds~$p$. The high-temperature paramagnetic (PM) phase at low temperatures gives way either to a ferromagnetic (FM) phase spin glass (SG) phase at weak and strong disorder respectively. \n For the dual model, we only indicated the schematic boundary of the FM phase.\n Note that although the temperatures~$T_c$ and~$T_c^*$ are related by the Kramers--Wannier relation, the dual model of the hyperbolic Ising model is different from the original model even on self-dual lattices (see main text for details). \n The phase boundary of the dual model corresponds to the decoding threshold of the hyperbolic surface code under phenomenological noise. The Nishimori line is indicated in dashed-gray. }\n \\label{fig:phases_sktech}\n\\end{figure}\n\nAt high temperature, both the RBIM and its dual are in the paramagnetic phase. As the temperature is lowered, at low disorder this gives way to a ferromagnetic phase which is continuously connected to that of the pure model at $p=0$. The transition from the paramagnet to the ferromagnet corresponds to an instability of the mean of the magnetization distribution $\\rho(m)$. That means while in the paramagnet we have \n\\begin{equation}\n \\rho(m) = \\delta(m),\n\\end{equation}\nin the ferromagnetic phase \n\\begin{equation}\n \\rho(m) = \\delta(\\abs{m - M}).\n\\end{equation}\n\nFor large disorder, $p \\approx 1\/2$, random systems can also develop spin glass order at low temperature, which corresponds to an instability in the variance of the magnetization distribution, which is also called the Edwards-Anderson (EA) order parameter \n\\begin{equation}\n q_{\\rm EA} = \\left[m^2\\right],\n\\end{equation}\nwhere the magnetization vanishes ($[m]=0$).\nAt intermediate values of disorder, there is in principle also the possibility of a magnetized spin glass phase \\cite{thouless1986, Carlson1990}, where the magnetization distribution has both finite width ($q_{\\rm EA} \\neq 0$) and mean ($[m] \\neq 0$). \n\nThe schematic phase diagram of the RBIM and its dual on the hyperbolic plane is shown in \\autoref{fig:phases_sktech}. Note that for the dual model, we only indicate the phase boundary of the ferromagnetic phase. There could exist a spin-glass phase in principle, but the investigation of that is beyond the scope of this work. \nWe also indicate the so called \\emph{Nishimori line} \\cite{Nishimori1981}, which is defined by the condition \n\\begin{equation}\n \\exp(2\\beta J) = \\frac{p}{1-p},\n \\label{eq:Nishimori}\n\\end{equation}\nthat is the (relative) probability of frustrating a bond due to thermal fluctuations is equal to that of flipping its sign due to the quenched disorder. \nIt is known that the multiciritical point in the RBIM lies on the Nishimori line and that the phase boundary of any magnetized phase must be reentrant or vertical, that is no magnetized phase can exist for $p_{\\rm N} < p$ \\cite{Nishimori1981}.\n\nAs indicated, we expect the ferromagnetic phase of the RBIM to have a larger extent than that of its dual, since the additional cocycle degrees of freedom $\\eta_{\\ell}$ have a finite contribution to the entropy, which is then strictly greater than that of the RBIM.\n\n\n\n\\section{Methods\\label{sec:methods}}\n\n\\subsection{High-Temperature Series Expansion\\label{sec:series_expansion}}\n\nOur primary means to map out the phase diagram of the random-bond Ising model in hyperbolic space will be to perform high-temperature series expansions of both the susceptibility \n\\begin{equation}\n \\chi = \\beta \\frac{1}{N} \\sum_{i,j} \\left[\\expval{\\sigma_i \\sigma_j} - \\expval{\\sigma_i}\\expval{\\sigma_j}\\right],\n\\end{equation}\nas well as of the Edwards-Anderson (EA) susceptibility\n\\begin{equation}\n \\chi_{\\rm EA} = \\beta \\frac{1}{N^2} \\sum_{i,j} \\left[\\expval{\\sigma_i \\sigma_j}^2 - \\expval{\\sigma_i}^2\\expval{\\sigma_j}^2\\right].\n\\end{equation}\nComing from a high-temperature, if there is a transition to low-temperatures ferromagnetic phase, the susceptibility $\\chi$ at the transition should diverge as a power law\n\\begin{equation}\n \\chi \\sim \\frac{1}{(T-T_c)^\\gamma}\n\\end{equation}\nwhile the Edwards-Anderson susceptiblity $\\chi_{\\rm EA}$ can have either a weak singularity or diverge as well \\cite{binder1986}. In contrast, if there is a transition into a low-temperature spin-glass phase, the susceptibility $\\chi$ will exhibit only a weak singularity (a cusp), while the Edwards-Anderson susceptibility diverges as a power law\n\\begin{equation}\n \\chi_{\\rm EA} \\sim \\frac{1}{(T-T_c)^{\\gamma'}}.\n\\end{equation}\n\n\\subsubsection{Biconnected graph expansion of inverse susceptibilities}\n\nIt turns out that for susceptibilties of the form \n\\begin{equation}\n \\chi_{k, l} = \\beta \\frac{1}{N} \\sum_{i,j} \\left[\\expval{\\sigma_i \\sigma_j}^k - \\expval{\\sigma_i}^k\\expval{\\sigma_j}^k\\right]^l,\n\\end{equation}\nit is favourable to perform the high-temperature expansion in the {\\em inverse} susceptibility. The reason for this is that it can be shown~\\cite{singh87} that the only non-trivial contributions come from \\emph{biconnected} graphs, that is graphs which stay connected if any of their vertices (and the edges attached to it) are being removed. \nWe show the first few graphs that contribute to the susceptibility $\\chi=\\chi_{1, 1}$ and EA-susceptibility $\\chi_{\\rm EA}=\\chi_{2, 1}$ on the $\\{5, 5\\}$ lattice in \\autoref{fig:55bicon}.\n\nThe inverse susceptibility can be expanded in terms of these graphs as a function of both inverse temperature $v=\\tanh(\\beta J)$ and disorder strength $\\mu = 1-2p$. In practice, the variables in the systematic biconnected graph expansion are $w=v^2$ and $\\alpha = \\mu\/v$:\n\\begin{align}\\label{eqn:sus_series}\n\\tilde{\\chi}^{-1}(w, \\alpha) = 1\\, + \\, \\sum_{g}\\, c(g)\\, W(g)\n\\end{align}\nwhere the sum is over all graphs, $c(g)$ is the coefficient of~$N$ of the number of embeddings of the graph~$g$ into the lattice and~$W(g)$ for each graph is a function of both~$w$ and~$\\alpha$. \nExpanding $W$ as a function of inverse temperature $w$, one can show that for each order $n$, the coefficient of $w^n$ is a polynomial in $\\alpha$ of order $n$ with integer coefficients. \nFor example, the inverse susceptibility on the $\\{5, 5\\}$ lattice is given by\n\\begin{align}\n\\chi^{-1}(w, \\alpha) = 1 &- 5 \\alpha w + 5 \\alpha^2 w^2 - 5 \\alpha^3 w^3 + 5 \\alpha^4 w^4 \\nonumber\\\\\n&+ (10 \\alpha + 10 \\alpha^2 +10 \\alpha^3 + 10 \\alpha^4 + 5 \\alpha^5) w^5 \\nonumber\\\\\n&+ \\order{w^6}.\n\\end{align}\nNote that for $\\alpha = 1$ (that is $v=\\mu$), we obtain the series on the Nishimori line up to order $\\order{w^n}=\\order{v^2n}$.\n\nFor more details and a derivation of \\autoref{eqn:sus_series} see Ref.~\\onlinecite{singh87}.\n\n\\begin{figure}\n\t{\\includegraphics[width=1.2cm]{55_bicon\/0.png}}\n\t{\\includegraphics[width=1.2cm]{55_bicon\/1.png}}\n\t{\\includegraphics[width=1.2cm]{55_bicon\/2.png}}\n\t{\\includegraphics[width=1.2cm]{55_bicon\/3.png}}\n\t{\\includegraphics[width=1.2cm]{55_bicon\/4.png}}\\\\\n\t{\\includegraphics[width=1.2cm]{55_bicon\/5.png}}\n\t{\\includegraphics[width=1.2cm]{55_bicon\/6.png}}\n\t{\\includegraphics[width=1.2cm]{55_bicon\/7.png}}\n\t{\\includegraphics[width=1.2cm]{55_bicon\/8.png}}\n\t{\\includegraphics[width=1.2cm]{55_bicon\/9.png}}\\\\\n\t{\\includegraphics[width=1.2cm]{55_bicon\/10.png}}\n\t{\\includegraphics[width=1.2cm]{55_bicon\/11.png}}\n\t{\\includegraphics[width=1.2cm]{55_bicon\/12.png}}\n\t{\\includegraphics[width=1.2cm]{55_bicon\/13.png}}\n\t{\\includegraphics[width=1.2cm]{55_bicon\/14.png}}\\\\\n\t\\caption{Some small biconnected subgraphs of the $\\{5,5\\}$-tiling. Removing a vertex and all its incident edges will leave the graphs connected. Only biconnected graphs contribute to the series expansion.}\n\t\\label{fig:55bicon}\n\\end{figure}\n\n\n\\subsubsection{Analysis of the series}\n\nWe analyze the generates series $\\tilde{\\chi}(w, \\alpha)$, usually for fixed $\\alpha$ as a function of $w$, using \\emph{first-order homogeneous integrated differential approximants (FO-IDAs)}.\nOne reason to choose FO-IDAs over simpler methods is that they are known to be less biased towards the lower-order coefficients of the expansion~\\cite{singh2}.\nThis is important, as the most relevant contributions on a $\\{r, s\\}$ tiling come from graphs with at least~$r$ edges.\n\nThe analysis using FO-IDAs proceeds as follows:\nFor fixed disorder strength $\\alpha$, we assume that the series $\\tilde{\\chi}$ is the solution of a first-order differential equation of the form\n\\begin{equation}\\label{eqn:IDAdef}\nQ_L(w) \\frac{d \\tilde{\\chi}(w)}{d v} + R_M(w)\\, \\tilde{\\chi}(w) + S_T(w) = 0\n\\end{equation}\nwhere $Q_L$, $R_M$ and $S_T$ are polynomials of degree $L$, $M$, $T$, respectively.\nBy equating the series order-by-order with the coefficients of \\autoref{eqn:IDAdef} we obtain a linear system of equations in the coefficients of the polynomials $Q_L$, $R_M$ and $S_T$.\nIt can be shown that for any root $w_c$ of the polynomial~$Q_L$, a solution of \\autoref{eqn:IDAdef} has an algebraic singularity of the form $(w-w_c)^{-\\gamma}$ \\cite{oitmaa2006}.\nThe exponent of the singularity is given by\n\\begin{align}\\label{eqn:crit_exp}\n\\gamma = \\frac{R_{M}(w_c)}{Q'_L(w_c)} .\n\\end{align}\nGenerally, the results for $w_c$ and $\\gamma$ will depend on the choice of degrees $L$, $M$ and $T$.\nIf we have the series up to order $N$ then we can choose all possible values satisfying $L+M+T \\leq N-2$.\nFollowing~\\cite{singh2} we exclude approximants if\n\\begin{itemize}\n\t\\item a root of $R_M$ is close to $w_c$, giving rise to a small estimate of $\\gamma$\n\t\\item a complex root of $Q_L$ with small absolute value smaller than $w_c$ is close to the real axis\n\\end{itemize}\n\nWe observe that the convergence of the series is very good, since the approximants for different choices of $L$, $M$ and $T$ are all close.\n\n\\subsection{Monte Carlo Simulations}\n\nTo corroborate our results from the series expansion and to compute additional observables, we also perform classical Monte-Carlo simulations for some sets of parameters. \nTo compute the disorder average $\\left[\\dots\\right]$, we perform Monte-Carlo simulations for using 1000 disorder realizations $\\left\\{J_{ij}\\right\\}$. For each realization, we simulate two independent copies $\\{ \\sigma_j^{(1)} \\}$, $\\{ \\sigma_j^{(2)} \\}$ of the system.\n\n\\subsubsection{Equilibration in the (possible) presence of glassiness}\n\nSince it is know that there is no spin glass behavior on the Nishimori line \\cite{Nishimori1981}, we expect that a standard local Metropolis-Hastings algorithm is sufficent to equilibrate the system at temperatures $T > 2J\\log[p\/(1-p)]^{-1}$. \nWhen approaching the spin glass phase, the local algorithm suffers from a drastic slowdown. Nevertheless, we are able to study the spin glass transition since for that we do not need to equilibrate the system deep inside the glassy phase. To make sure that the system is actually equilibrated, we keep track of the autocorrelation time of all relevant observables (computed via binning analysis~\\cite{ALPSCore}) to ensure that we equilbrate the system for at least $10$ times as long as the largest equilibration time in the system and that we take $5000$ independent samples per temperature value for each observable\n\n\\subsubsection{Finite size scaling}\nDue to the absence of a unique linear dimension in the compactifications of the hyperbolic plane, we perform finite size scaling as a function of the number of sites~$N$. This was initially proposed for a fully connected model~\\cite{Botet1982} and has been used for hyperbolic lattices with open boundary conditions \\cite{Shima2006} as well as in our study of the pure Ising model in the hyperbolic plane \\cite{breuckmann2020}.\nThe main idea is that a quantity $A$, close to criticality, follows a scaling form\n\\begin{equation}\n A \\sim \\abs{T-T_c}^a \\, F\\left(N\/N_c\\right)\n\\end{equation}\nwith a correlation number $N_c$.\nAssuming that a corresponding system of finite dimension $d = d_c$, where $d_c$ is the upper critical dimension, has the same scaling behavior as its hyperbolic sibling, it follows that \n\\begin{equation}\n N_c = \\sim \\abs{T-T_c}^{-\\mu},\n\\end{equation}\nwith the critical exponent \n\\begin{equation}\n \\mu = \\nu_{\\rm MF} \\, d_c,\n\\end{equation}\nand $\\nu_{\\rm MF}$ is the mean-field value of the critical exponent of the correlation length $\\xi$.\n\n\\subsubsection{Observables}\n\nTo map out the phase diagram and compute critical exponents, we study a number of observables, all of which are related to either the magnetization\n\\begin{align}\n m^{(\\alpha)} = \\frac{1}{N} \\sum_{j} \\sigma_j^{(\\alpha)} \\label{eq:mag}\n\\end{align}\nor the Edwards-Anderson order parameter\n\\begin{equation}\n q = \\frac{1}{N} \\sum_{j} \\sigma_j^{(1)} \\sigma_j^{(2)}.\n\\end{equation}\nFirst, to determine the location of the critical point and the critical exponent of the correlation number $\\mu$, we compute the binder cumulants\n\\begin{align}\n g &= 1 - \\frac{\\left[\\expval{m^4}\\right]}{\\left[\\expval{m^2}^2\\right]}, \\label{eq:binder}\\\\\n g_{\\rm EA} &= 1 - \\frac{\\left[\\expval{q^4}\\right]}{\\left[\\expval{q^2}^2\\right]} \\label{eq:binderEA}\n\\end{align}\nwhich, for different system sizes, cross at the transition to a magnetized and a spin glass phase respectively. The best estimate for the transition temperature $T_c$ and the exponent $\\mu$ is given by performing a data collapse, using the fact that close to the transition the respective cumulant is given by\n\\begin{equation}\n g = G\\left(N^{1\/\\mu}(T-T_c)\\right),\n\\end{equation}\nwith some universal scaling function $G$.\n\nFor both order parameters, we also compute the corresponding susceptibilties \n\\begin{align}\n \\chi &= \\beta N\\left( \\left[\\expval{m^2}\\right] - \\left[\\expval{m}\\right] \\right), \\label{eq:sus}\\\\\n \\chi_{\\rm EA} &= \\beta N\\left( \\left[\\expval{q^2}\\right] - \\left[\\expval{q}\\right] \\right). \\label{eq:susEA}\n\\end{align}\nAgain, the best estimate for $T_c$, $\\gamma$ and $\\mu$ are obtained by performing a data collapse, since close to the transition the susceptibility is given by\n\\begin{equation}\n \\chi = N^{\\gamma\/\\mu} S\\left(N^{1\/\\mu}(T-T_c)\\right),\n\\end{equation}\nwith some universal scaling function $S$.\n\n\n\\section{Results for the RBIM\\label{sec:rbim}}\n\n\\subsection{Phase diagram on the \\{5, 5\\} lattice}\n\nTo study general features of the phase diagram of the RBIM in hyperbolic space as well as to assess the reliability of the high-temperature series expansion (HTSE) in the presence of disorder, we first map out the phase diagram of the model on the $\\{5, 5\\}$ lattice in detail, using both HTSE as well as Monte-Carlo simulations.\n\n\\begin{figure}\n \\centering\n \\includegraphics{phases_55.pdf}\n \\caption{Phase diagram on the $\\{5, 5\\}$ lattice as a function of temperature $T$ and disorder strength $p$. We show both the magnetization $m$ as well as the Edwards-Anderson order parameter $q_{\\rm EA}$ (inset) obtained from Monte-Carlo (MC) simulations of a $N=1920$ system. We superimpose this with the phase boundaries obtained from the high-temperature series expansion (HTSE) and MC (see main text for details).}\n \\label{fig:phases55}\n\\end{figure}\n\nThe phase diagram of $\\{5, 5\\}$ is obtained from HTSE and MC simulations is shown in \\autoref{fig:phases55}. Compared to the RBIM on the euclidean square ($\\{4, 4\\}$) lattice, we find a much larger ferromagnetic phase and a extended spin glass phase. In contrast to the Bethe lattice, here we do not find evidence for a magnetized spin glass phase, although our low-temperature data is not good enough to rule out a very small extent.\n\nTurning to explain our results in more detail, in \\autoref{fig:phases55} we show both, the magnetization $m$ as well as the Edwards-Anderson order parameter $q$ (in the inset) as obtained from a MC simulation with system size $N = 1920$. While the magnetization is nonzero only in the ferromagnetic phase, the EA order parameter is nonzero in both the ferromagnet and the spin glass. \nWe superimpose these plots with the critical points obtained using finite-size scaling of the MC data (open circles) and with the critical lines obtained from HTSE of the (EA-) susceptibility (solid lines). In both methods, we can distinguish the transition from the paramagnet to a ferromagnetic phase and that to a spin glass phase reliably. In the finite size analysis of the MC data, a transition to the ferromagnetic phase is signaled by a crossing of both the binder cumulant of the magnetization, $g$ [\\autoref{eq:binder}] as well as a crossing of the binder cumulant of the Edwards-Anderson order parameter, $g_{\\rm EA}$ [\\autoref{eq:binderEA}]. In contrast, at the transition to a spin glass phase, only $g_{\\rm EA}$ shows a crossing while $g$ does not, since the magnetization $m$ vanishes in the spin glass. Finite size scaling along the Nishimori line indicates a transition at $p_{\\rm N} = 0.247 \\pm 0.02$ and finite size analysis as a function of temperature at constant disorder shown a transition into a ferromagnet for $p \\lessapprox p_{\\rm N}$ and a transition into a spin glass for $p \\gtrapprox p_{\\rm N}$, making the Nishimori point the multicritical point. \n\nThis result is corroborated by HTSE analysis. Here, a transition to the ferromagnet (spin glass) is signaled by the divergence ferromagnetic (EA-) susceptibility $\\chi_{(\\rm EA)}$. Note that since the non-divergent susceptibility at both transitions typically also has a weak singularity (a cusp), series analysis normally predicts a divergence for both susceptibilities, but at different critical temperatures. In practice, we distinguish the two transitions by the fact which susceptibility is predicted to diverge at a larger temperature. Along the Nishimori line, that is $\\alpha = 1$ in \\autoref{eqn:sus_series}, the two susceptibilities are equal and HTSE yields a critical point $w_c = 0.256456 \\pm \\num{8.6e-6}$, which corresponds to $p_{\\rm N} = 0.246793 \\pm \\num{4.2e-6}$. For $\\alpha < 1$ we find a transition to a ferromagnetic phase while for $\\alpha > 1$ we find a transition into a spin glass phase, again suggesting that the Nishimori point is indeed the multicritical point of the model.\n\n\n\\subsection{Phase boundaries for different tilings: coordination vs curvature}\n\n\\begin{figure}\n \\centering\n \\includegraphics[width=\\columnwidth]{phases_curvature.pdf}\n \\caption{Critical temperature $T_c$ obtained from high-temperature expansion, for different tilings of the hyperbolic plane. The inset shows $v_c = \\tanh(J\/T_c)$ as a function of curvature $\\kappa$ for the pure model ($p=0$), along the Nishimori line ($(1-p)\/p = e^{-\\beta J}$) and for the spin glass boundary ($p=1\/2$).}\n \\label{fig:phases_curv}\n\\end{figure}\n\nWe now use the high-temperature expansion to study how the paramagnet-ferromagnet and paramagnet-spin-glass phase boundaries vary for different tilings $\\{r, s\\}$. For low disorder, the critical temperature is mostly controlled by the coordination number $s$ and for $p=0$ even agrees quantitatively with that of the Bethe lattice with the same coordination \\cite{breuckmann2020}.\nQualitatively, this behaviour can be understood by considering that the transition into the ferromagnet at low disorder is driven by a competition between and the entropy of the paramagnet and internal energy of the ferromagnetic state\n\\begin{equation}\n E_{\\rm FM} = \\frac{sN}{2}\\left[J_{ij}\\right],\n\\end{equation}\nwhich is proportional to the coordination number $s$. This means that with larger~$s$, the ferromagnet becomes more favorable at larger temperatures. \nAs disorder is increased however, $[J_{ij}]$ also increases (approaching zero from a negative value) and so does the importance of~$s$ as a control parameter for the transition temperature. \nFinally, $[J_{ij}] \\to 0$ as $p\\to \\frac{1}{2}$ and the critical temperature becomes a monotonic function of the curvature $\\kappa$, as seen in the inset of \\autoref{fig:phases_curv}.\n\n\\subsection{Critical Behaviour}\n\n\\begin{table}\n \\centering\n \\begin{tabular}{c|c|c|c}\n & $p=0$ & Nishimori Line & $p=1\/2$\\\\\\hline\n $\\mu$ & 2 & $3.0 \\pm 0.1$ & $2.0 \\pm 0.1$ \\\\\n \n $\\gamma$ & $1.000001 \\pm 0.000005$ & $1.0003 \\pm 0.0008$ & -\\\\\n $\\gamma_{\\rm EA}$ & - & $1.0003 \\pm 0.0008$ & $1.0011 \\pm 0.0025$ \\\\\n $\\beta$ & $0.46 \\pm 0.05$ & $1.00 \\pm 0.05$ & -\n \\end{tabular}\n \\caption{Critical exponents on the $\\{5, 5\\}$ lattice along different scaling axes. We estimate the correlation volume exponent, $\\mu$, from finite size analysis of the binder parameter $g$. For the susceptibility exponents $\\gamma$ and $\\gamma_{\\rm EA}$ the best estimates are obtained via HTSE analysis. }\n \\label{tab:exponents}\n\\end{table}\n\n\\begin{figure}\n \\centering\n \\includegraphics{collapse_nishimori}\n \\caption{Finite size scaling collapse of the Binder cumulant~$g$ [\\autoref{eq:binder}], the susceptibility [\\autoref{eq:sus}], and the magnetization $m$ [\\autoref{eq:mag}] of the random bond Ising model on the $\\{5, 5\\}$ lattice along the Nishimori line [\\autoref{eq:Nishimori}].}\n \\label{fig:collapse-N}\n\\end{figure}\n\nIn \\autoref{tab:exponents}, we show out best results for the critical exponents for the $\\{5, 5\\}$ lattice for different scaling axis (with the $p=0$ results taken from Ref. \\onlinecite{breuckmann2020}).\nThe best results are typically obtained from the HTSE, except for the exponent $\\mu$ of the correlation volume, which we compute by finite size analysis of the Monte-Carlo data. \nIn all cases were results from both methods are available, they are compatible within errors. \nThe best finite-size scaling collapse of the Monte Carlo data along the Nishimori line is shown in \\autoref{fig:collapse-N}.\nThe best collapse is obtained for slightly different values of $p_c$ for the susceptibility and the binder cumulant, which we attribute to finite size effects.\n\nThe results in \\autoref{tab:exponents} are all compatible with the mean-field expectation, except for the exponents $\\mu = 3$ and $\\beta = 1$, observed along the Nishimori line. This is because as established in \\autoref{sec:rbim}, the Nishimori line passes through the multicritical point, which generally shows distinct critical behavior even in (effectively) infinite dimensions. \nNote that still, the exponents are consistent with the hyperscaling relation\n\\begin{equation}\n \\mu = 2\\beta + \\gamma\n\\end{equation}\nNote that the specific heat does not develop a power-law singularity for any of the transitions considered and hence we do not present a critical exponent $\\alpha$.\n\n\\section{Results for the dual-RBIM\\label{sec:dual-rbim}}\n\nIn this section, we present results of Monte-Carlo simulations of the dual random-bond Ising model (dual-RBIM). We present strong evidence that this model exhibits a strongly first-order transition as a results of its cocycle degrees of freedom and numerically verify that for $p=0$, the critical temperature of this transition is indeed the Kramers--Wannier dual to the critical temperature of the Ising model on the dual lattice.\n\n\\subsection{Dual Ising model}\n\n\\begin{figure}\n \\centering\n \\includegraphics{monte_carlo_dual_p=0}\n \\caption{Evidence for a strongly first-order phase transition of the pure dual Ising model (that is \\autoref{eq:H-dual-ising} with $p=0$) on the $\\{5, 5\\}$ lattice. We show the vertex magnetization $m = \\expval{\\sigma_j}$, its Binder cumulant $g$ as well as the loop magnetization $m_{\\eta} = \\expval{\\eta_\\ell}$ and its Binder cumulant $g_{\\eta}$ as a function of temperature.}\n \\label{fig:dual-mc}\n\\end{figure}\n\nIn \\autoref{fig:dual-mc}, we show results from Monte-Carlo siumulations of the dual Ising model, that is \\autoref{eq:H-dual-ising} with $p=0$, on the $\\{5, 5\\}$ lattice. We show the average vertex magnetization $m = \\expval{\\sigma_j}$ and \\emph{loop} magnetization $m_{\\eta} = \\expval{\\eta_\\ell}$, as well as the Binder cumulants $g$ and $g_\\eta$ for vertex and loop magnetization respectively. The fact that the magnetizations for different system sizes cross at a single point, together with the pronounced dip of the Binder cumulants just before the transition are strong evidence that both quantities undergo a strongly first-order transition.\n\nSince the $\\{5, 5\\}$ lattice is self dual and the Kramers--Wannier duality [\\autoref{eq:krammers-wannier-T}] is exact at $p=0$, we expect the transition to occur at a critical temperature dual to the the critical point of the Ising model. Subsituting $T_c = 3.93$ \\cite{breuckmann2020} into \\autoref{eq:krammers-wannier-T} yields $T_{\\rm c}^* \\approx 1.44$, which we indicate in \\autoref{fig:dual-mc} by a vertical dashed line and is in good agreement with the position of the crossing of both Binder cumulants and magnetizations.\n\n\\begin{figure}\n \\centering\n \\includegraphics{wang_landau_p=0}\n \\caption{Free energy difference [\\autoref{eq:fdiff}] between the Ising model and its dual on the $\\{5, 5\\}$ lattice.}\n \\label{fig:dual-wl-pure}\n\\end{figure}\n\nTo corraborate the above findings, we also implement the Wang-Landau algorithm \\cite{wang_landau2001, wang_landau2001b, schulz2003, belardinelli2007} and compute the free energy difference of the Ising model and its dual that is\n\\begin{equation}\n \\label{eq:fdiff}\n \\Delta F(T) = \\log[Z_{\\rm tot}(T)] - \\log[Z_{0}(T)].\n\\end{equation}\nHere, $Z_{\\rm tot}$ is the partition function of the dual Ising model, that is it includes a sum over all cocycle variables~$\\eta_\\ell$ (therefore the subscript `tot'). $Z_0$ is the partition function of the Ising model on the same lattice, that is we fix $\\eta_\\ell = 1$ for all $\\ell$.\nBecause of the latter relation between $Z_{\\rm tot}$ and $Z_0$, we have $\\Delta F > 0$ for all~$T$. In the ordered phase of the dual Ising model, the difference vanishes since the sum over cocycle variables does not contribute. This is shown in \\autoref{fig:dual-wl-pure}. The quantity $\\Delta F$ also has the advantage of indicating both phase transitions in one observable, since the free energy of the Ising model shows a visible kink at $T_{\\rm c}$. Both critical temperatures are again indicated in the figure by vertical dashed lines.\n\n\\subsection{Dual random bond Ising model} \n\n\\begin{figure}\n \\centering\n \\includegraphics{wang_landau_Nishimori}\n \\caption{Free energy difference [\\autoref{eq:fdiff}] between the random bond Ising model (RBIM) and the dual-RBIM on the $\\{5, 5\\}$ lattice along the Nishimori line. The shaded region indicates the location of the Nishimori point $p_{\\rm N} = 0.0228 \\pm 0.0010$. The inset shows the best data collapse, assuming the same correlation exponent $\\mu$ as in the RBIM.}\n \\label{fig:dual-wl-Nishimori}\n\\end{figure}\n\nIn the case of the random model, the dual-RBIM is not exactly dual to the RBIM and hence we have a a-priori guess for the location of the critical point. Additionally, as is already the case in the pure model, the strongly first order nature of the transition complicates its numerical investigation. We find that single-spin flip Monte-Carlo is unreliable even for small system sizes. However the Wang-Landau algorithm is still converging and hence we can infer the location of the critical point from the free energy difference [\\autoref{eq:fdiff}]. The difference for $T < T_c^*$ vanishes as a function of system size and diverges as a function of system size for $T_c^* < T$ respectively. \nIn \\autoref{fig:dual-wl-Nishimori}, we show $\\Delta F$ as a function of disorder strength~$p$ along the Nishimori line [\\autoref{eq:Nishimori}]. The data is consistent with a transition at $p_{\\rm N}^* = 0.0228 \\pm 0.001$, which is indicated in the figure by a shaded area. The inset shows the best data collapse assuming the same correlation exponent $\\mu = 3$ as in the RBIM.\n\nSubstituting the value of $p_{\\rm N} = 0.246793 \\pm \\num{4.2e-6}$ obtained from the high-temperature expansion of the RBIM (see \\autoref{sec:rbim} for details) into the duality relation conjectured by Nishimori (\\autoref{eq:duality-conjecture}) and solving for $p_{\\rm N}^*$ yields a value of $p_{\\rm N}^* = 0.029891 \\pm \\num{2e-6}$. As observed for the RBIM on a range of euclidean lattice geometries \\cite{takeda2005exact} this is somewhat close to our numerical result but not compatible within error bars.\n\n\n\n\\section{Quantum Error Correction\\label{sec:qec}}\n\nQuantum error correcting codes are used in quantum computation to reduce the effects of decoherence.\nCertain infinite families of codes, together with associated quantum error correction protocols, can be shown to have a \\emph{threshold}.\nA threshold is a critical value of a noise parameter, below which the error correction protocol succeeds with probability approaching 1 with increasing code sizes.\n\nIt was argued in \\cite{dennis2002topological,wang2003confinement} that the threshold of the toric code corresponds to the phase transition point along the Nishimori line of the RBIM on the square-grid $\\{4,4\\}$.\nIn~\\cite{kubica2018three} it was proved that this is indeed the case for quantum codes which encode a finite number of qubits.\nIn~\\cite[Section~IV-C]{chubb2021statistical} it was mentioned that the statistical mechanical models associated to quantum codes which encode an extensive number of qubits may exhibit multiple phase transitions.\nThis behaviour was studied in \\cite{kovalev2018numerical}.\n\nThe quantum codes associated to the hyperbolic RBIM are called \\emph{hyperbolic surface codes}~\\cite{breuckmann2016constructions,breuckmann2017hyperbolic,conrad2018small}.\nThese codes do encode an extensive number of qubits, so that the proofs of \\cite{kubica2018three,chubb2021statistical} do not apply to them.\nIn \\cite{jiang2019duality} the authors consider the hyperbolic RBIM and give a condition sufficient for error correction to be possible, which is equivalent to $\\Delta F \\to 0$, where $\\Delta F$ is the free energy difference of the RBIM and the dual-RBIM, see \\autoref{eq:fdiff}. \nHence, the phase transition of what we call the ``dual-RBIM'' along the Nishimori line corresponds exactly to the maximum likelihood decoding threshold of the hyperbolic surface code under independent bit- and phase-flip noise\n\\begin{equation}\n p_{\\rm th, ML} = p_{\\rm N}^* = 0.0228 \\pm 0.0010.\n\\end{equation}\nThis can be compared to the threshold when using a minimum-weight perfect-matching decoder, which is $p_{\\rm th, MWPM} \\approx 0.0175$ \\cite{breuckmann2017homological}.\nUsing an optimal decoder rather than MWPM hence increases the threshold by about 27\\%.\n\n\n\n\n\\section{Conclusion\\label{sec:conclusion}}\n\nTo summarize, we have presented an in-depth study of the random bond Ising model (RBIM) on the hyperbolic plane as well as the model that is its Kramers--Wannier dual in the absence of disorder. Resolving a conundrum raised in earlier work \\cite{Rietman1992, breuckmann2020}, we showed that this ``dual-RBIM'' is different from the RBIM even on self-dual lattices due to the extensive number of nontrivial cocycles of hyperbolic lattices. Combining high-temperature expansion techniques and Monte-Carlo techniques, we mapped out the phase diagrams of both models, establishing the existence of a spin-glass phase with the Nshimori point as the tricritical point. Studying the critical properties of the high-temperature transitions, we showed that with the exception of the multicritical point, all transitions are mean-field in nature.\nWe verified the duality of both models explicitly in the disorder-free case and showed that the extended duality as conjectured by Takeda, Sesamoto and Nishimori \\cite{takeda2005exact} is fulfilled only approximately.\nFinally, we commented on the relation of the above findings to the decoding of hyperbolic surface codes and argued that the critical disorder along the Nishimori of what we call the dual-RBIM corresponds to the maximum-likelihood decoding threshold of hyperbolic surface codes under independent bit- and phase-flip noise. This generalizes the statistical mechanics mappings of the decoding of zero-rate quantum codes\\cite{dennis2002topological, chubb2021statistical, kubica2018three} to quantum codes with \\emph{finite} rate.\n\nThis work open up multiple interesting ares for future work. For example, beyond the scope of the current paper was a detailed investigation of the nature of the spin-glass phase in hyperbolic space and in particular its fate in the dual-RBIM. \nMoreover, a detailed investigation of the phase space structure of the dual model could yield valuable insights into the decoding of finite-rate quantum codes.\n\n\n\\subsection*{Acknowledgements}\nWe thank Leonid Pryadko, Aleksander Kubica, Sounak Biswas, Rajiv Singh and Roderich Moessner for helpful discussions, and also Philippe Suchsland and Dmitry L. Kovrizhin and Peng Rao for helpful comments on the manuscript.\nBP acknowledges support by the Deutsche Forschungsgemeinschaft under grants SFB 1143 (project-id 247310070) and the cluster of excellence ct.qmat (EXC 2147, project-id 390858490). \nNPB acknowledges support through the EPSRC Prosperity Partnership in Quantum Software for Simulation and Modelling (EP\/S005021\/1).\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}