data_all_eng_slimpj/shuffled/split2/finalzzaett

{"text":"\\section{Introduction}\n\nIn a fascinating paper \\cite{Cachazo:2013hca}, Cachazo, He and Yuan constructed a way to write the $n$-point amplitudes for Yang-Mills and for gravity in $D$ dimensions. They wrote the amplitudes as $n$-dimensional integrals on the support of the so-called scattering equations. Later on, the formalism was shown to be well-suited to describe other theories as well, such as bi-adjoint scalars \\cite{Cachazo:2013iea}, Dirac-Born-Infeld and the non-linear sigma model \\cite{Cachazo:2014xea}. These compact formulae were subsequently shown to also arise from ambitwistor strings \\cite{Mason:2013sva,Casali:2015vta}.\n\n\\begin{table}\n\\begin{center}\n\\begin{tabular}{|c|c|c|c|c|}\n\\hline\n\\backslashbox{\\begin{tabular}{l}Left\\\\Action\\end{tabular}}{\\begin{tabular}{r}Right\\\\Action\\end{tabular}}&\\begin{tabular}{c}Current\\\\Algebra\\end{tabular}&\\begin{tabular}{c}Single\\\\Fermion\\end{tabular}&\\begin{tabular}{c}Two\\\\Fermions\\end{tabular}&None\\\\\n\\hline\n\\begin{tabular}{c}Current\\\\Algebra\\end{tabular}&\\begin{tabular}{c}Bi-adjoint\\\\scalar\\end{tabular}&Yang-Mills&\\begin{tabular}{c}Non-linear\\\\sigma model\\end{tabular}&$(DF)^2$\\\\\n\\hline\n\\begin{tabular}{c}Single\\\\Fermion\\end{tabular}&Yang-Mills&\\begin{tabular}{c}Einstein\\\\Gravity\\end{tabular}&Born-Infeld&\\begin{tabular}{c}Conformal\\\\Gravity\\end{tabular}\\\\\n\\hline\n\\begin{tabular}{c}Two\\\\Fermions\\end{tabular}&\\begin{tabular}{c}Non-linear\\\\sigma model\\end{tabular}&Born-Infeld&Galileon&$(DF)^2$-photon\\\\\n\\hline\nNone&$(DF)^2$&\\begin{tabular}{c}Conformal\\\\Gravity\\end{tabular}&$(DF)^2$-photon&$\\rm(Weyl)^3$\\\\\n\\hline\n\\end{tabular}\n\\end{center}\n\\caption{The matrix of ambitwistor string actions with the new row\/column (None).\\label{teorimatrix}}\n\\end{table}\n\n\nIn this paper we will add three extra theories to the list of those that admit a simple CHY-type formulation. The first theory is the $(DF)^2$ theory constructed in \\cite{conformal}. This theory is related to conformal gravity \\cite{Fradkin:1985am} via the KLT relations \\cite{Kawai:1985xq}. We compute its lower point amplitudes and subsequently find an $n$-point generalization, that possesses the correct factorization channels. The CHY formulation makes the absence of all $\\varepsilon_i\\cdot\\varepsilon_j$-terms in the amplitudes manifest, a property that is otherwise obscure from a Feynman diagram representation.\n\nThe second theory we consider is conformal gravity itself\\footnote{or more accurately a $D$-dimensional $R^2$ theory which in $D=4$ becomes conformal gravity. Throughout the paper we will use these terms interchangeably. There are other types of $R^2$ gravity but this is the only one of interest to us}. For this theory we also propose a CHY formulation for the $n$-point amplitude and show that it factorizes correctly. The formula beautifully generalizes the one by Berkovits and Witten for conformal gravity \\cite{Berkovits:2004jj}, to which it reduces when considering the MHV sector in $D=4$.\n\n\n\n\n\n\nFinally, we show that these theories can be given a straightforward interpretation in terms of ambitwistor strings. In our investigation of the corresponding ambitwistor string theories, we find a third theory which can be given a simple CHY formulation. This theory consists of a photon field governed by a $(DF)^2$ term and coupled to Einstein gravity. With these theories in hand, we can expand the usual matrix of possible ambitwistor theories with a new row\/column. The new matrix of ambitwistor theories is shown in table \\ref{teorimatrix}, with different choices of ambitwistor actions and the resulting theories coming from these actions. Note that the (Weyl)${}^3$-theory is just the usual bosonic ambitwistor string, corresponding to the choice (None, None).\n\n\n\n\n\n\nAt tree level, the theories we analyze can also be interpreted as sectors of previously considered ambi\\-twistor models. For example, the conformal gravity given by the (Single Fermion, None)-choice is a sector of the heterotic ambitwistor string given by (Single Fermion, Current Algebra), in the same sense that Berkovits--Witten is a sector of Witten's twistor string \\cite{Witten:2003nn} . In fact, the same is true for any pair of theories of the form \\{(X, None); (X, Current Algebra)\\}.\nNonetheless, it is remarkable that the ambitwistor approach allows us to truncate the larger models and consider those sectors themselves as stand-alone theories,\nand the applicability of this is exemplified by the fact that the theories considered in this paper had not been discussed before in the context of ambitwistor strings.\n\n\n\n\nOne should note that the theories studied in this paper are un-physical, due to the presence of modes with a wrong-sign propagator which render the theories non-unitary. However, they are interesting to study because of their relationships with well-known, physical theories. Conformal supergravity can be related to Einstein gravity in asymptotically (anti-) de Sitter space \\cite{Maldacena:2011mk} and its $U(1)$ anomaly can be used to study the similar anomaly in Poincar\\'e supergravity \\cite{Carrasco:2013ypa}.\n Furthermore, the $\\alpha^\\prime \\to \\infty$ limit of the heterotic string should also be related to some kind of conformal gravity. The theory with a $(DF)^2$ photon coupled to Einstein gravity, which we mentioned above and will describe further on in section \\ref{ambitwist sec}, is also related to a physical theory. By taking a specific limit, it is possible (at tree level) to relate the amplitudes of the photons to graviton amplitudes from pure Einstein gravity. This provides an alternative route for generating (tree-level) gravity amplitudes through the double copy by merging the $(DF)^2$ theory of Johansson and Nohle with the non-linear sigma model. As for the $(DF)^2$ itself, apart from being a piece in the double-copy constructions, it is of interest for the ambitwistor string community since it helps clarify some aspects of the theory, as we show in this paper.\n\nThe paper is structured as follows. We will begin by describing some basic properties of gluon amplitudes, the $(DF)^2$ theory and the similarities between the amplitudes of this theory and those of Yang-Mills (section \\ref{DFtheory}). Then we will review the scattering equations and the CHY-formulation of amplitudes, as well as some functions that will prove useful later (section \\ref{scat eq sec}). We will then argue that the theories in question give rise to amplitudes that are extremely simple when written in the CHY-formulation (section \\ref{amp sec}). Subsequently we show how these simple formulae can arise from ambitwistor theories (section \\ref{ambitwist sec}). Finally we sum up our results in the conclusions.\n\n\\section{\\boldmath{$(DF)^2$ Theory}}\\label{DFtheory}\n\nThe $(DF)^2$ theory created by Johansson and Nohle \\cite{conformal} will play an essential role in this paper so in this section the theory will briefly be described. We should perhaps note that Lagrangians with similar operators have previously been studied for phenomenological reasons in \\cite{Simmons:1989zs,Simmons:1990dh,Cho:1993eu,Duff:1991ad,Dreiner:1991xi,Dixon:1993xd} and because the operators arise as corrections in the $\\alpha'$ expansion of bosonic open string theory \\cite{Barreiro:2012aw,Barreiro:2013dpa,Boels:2016xhc}. It is however the specific theory introduced in \\cite{conformal} that interests us as it satisfies color-kinematics duality and gives conformal gravity through the double copy. In general the amplitudes of this theory have many features similar or identical to the beautiful features of Yang-Mills amplitudes. For this reason it will be useful to review some of the basic properties of Yang-Mills amplitudes.\n\n\n\nFor starters the tree-level amplitudes of gluons in Yang-Mills theory can be written as a sum over single-trace color factors and corresponding color-ordered amplitudes:\n\n\\begin{align}\n{\\cal A}_n^{\\rm tree}&=g^{n-2}\\sum_{{\\rm perm} (2,3,\\ldots n)} {\\rm Tr}(T^{a_1} T^{a_2} T^{a_3} \\cdots T^{a_n}) \\, A(1,2,3, \\ldots,n).\\label{c-o amp}\n\\end{align}\n\nUsing the Kleiss-Kuijf relations \\cite{Kleiss:1988ne}, this can be re-expressed as a sum over strings of structure constants:\n\n\n\\begin{align}\n{\\cal A}_n^{\\rm tree}&=(ig)^{n-2}\\sum_{{\\rm perm} (2,3,\\ldots n-1)} f^{a_1a_2b_1}f^{b_1a_3b_2}\\cdots f^{b_{n-3}a_{n-1}a_n} \\, A(1,2,3, \\ldots,n),\n\\end{align}\n\n\\noindent where the color-ordered amplitudes are the same as in \\eqref{c-o amp}. This is known as the DDM basis \\cite{DelDuca:1999iql,DelDuca:1999rs}, and it is the form that the amplitudes from the ambitwistor string naturally appear in.\n\nThe gluon amplitudes of Yang-Mills are also known to satisfy the color-kinematics duality \\cite{Bern:2008qj} (see also \\cite{Sondergaard:2009za}), which works as follows. Consider an $n$-point amplitude written in the form:\n\n\\begin{align}\n\\mathcal{A}_n=&{}(ig)^{n-2}\\sum_{i\\ \\in\\ \\textrm{cubic graphs}}\\frac{n_i c_i}{D_i},\\label{amp for c-k}\n\\end{align}\n\n\\noindent where the $c_i$'s are products of structure constants, the $n_i$'s are kinematic numerators and the $D_i$'s are products of propagators. There is a certain ambiguity in how the numerators are chosen because the $c_i$'s are dependent on each other due to the Jacobi relations. However the color-kinematics duality tells us that it is possible to chose the numerators in such a way that they satisfy relations identical to the Jacobi relations for the corresponding color factors.\n\nFor the color-ordered amplitudes the duality leads to the BCJ relations \\cite{Bern:2008qj} (proven from a string theory perspective in \\cite{BjerrumBohr:2009rd} and from a field theory perspective in \\cite{Feng:2010my} using the BCFW recursion relations \\cite{Britto:2004ap,Britto:2005fq}).\n\n\n\\begin{align}\n0=p_1\\cdot p_2 A(1,2,3, \\ldots,n)+\\sum_{i=3}^{n-1} (p_1\\cdot p_2 + p_2\\cdot p_3+\\cdots +p_2\\cdot p_i) A(1,3,\\cdots, i,2,i+1, \\cdots,n).\\label{BCJ amp}\n\\end{align}\n\nWriting the amplitudes in a form satisfying color-kinematics duality has the advantage that it makes the relationship between Yang-Mills and Einstein gravity straightforward. If the numerators satisfy the duality, one simply replaces the color factors, $c_i$, by another copy of the numerators, $n_i$, in order to arrive at the amplitudes for gravity. This is known as the double copy and is equivalent to the KLT relations:\n\n\\begin{align}\n{\\cal M}_n^{\\rm EG}&= A^{\\rm YM} \\cdot S \\cdot A^{\\rm YM}\\,,\\label{KLT EG}\n\\end{align}\n\n\\noindent where the color-ordered gauge-theory amplitudes have been packaged into column\/row vectors of $(n-3)!$ size, and the matrix $S$ is the (field theory) KLT kernel.\n\nSchematically we can write this as:\n\n\\begin{align}\n{\\rm EG} &= {\\rm YM} \\otimes {\\rm YM}\\,.\n\\end{align}\n\nThese are the properties of Yang-Mills theory that will be relevant for our discussion of the $(DF)^2$ theory which we will now turn to. The Lagrangian of this theory is given by:\n\n\n\n\\begin{align}\n{\\cal L}_{(DF)^2}=&{}   \\frac{1}{6}(D_{\\mu} F^{a\\, \\mu \\nu})(D^{\\rho} F^{a}_{\\phantom{a}\\, \\rho \\nu}) + \\frac{1}{3} (D^{\\rho} F^{a\\, \\mu \\nu})(D_{\\mu} F^{a}_{\\phantom{a}\\, \\rho \\nu})   + \\frac{1}{2}g \\,  C^{\\alpha ab}  \\varphi^{ \\alpha}   F_{\\mu \\nu}^a F^{b\\, \\mu \\nu }  \\\\\n&+\\frac{1}{2}(D_{\\mu} \\varphi^{\\alpha})^2+ \\frac{1}{3!} g \\, d^{\\alpha \\beta \\gamma}   \\varphi^{ \\alpha}  \\varphi^{ \\beta} \\varphi^{ \\gamma}\\nn.\n\\end{align}\nwhere the field strength and the covariant derivatives are defined as\n\\begin{align}\nF_{\\mu \\nu}^a &= \\partial_{\\mu} A_{\\nu}^a-\\partial_{\\nu} A_{\\mu}^a + g  f^{abc} A_{\\mu}^b A_{\\nu}^c, \\nn \\\\\nD_{\\rho} F_{\\mu \\nu}^a &= \\partial_{\\rho} F_{\\mu \\nu}^a +  g  f^{abc} A_{\\rho}^b F_{\\mu \\nu}^c  , \\label{fieldDef}\\\\\nD_{\\mu} \\varphi^\\alpha  &= \\partial_{\\mu} \\varphi^\\alpha -  i g  (T^{a})^{\\alpha \\beta} A_{\\mu}^a \\varphi^\\beta  \\nn.\n\\end{align}\n\nThe scalar $\\varphi^\\alpha$ transforms in a real representation of the gauge group, with generator $(T^a)^{\\alpha \\beta}$. Some of the interactions are parametrized by symmetric Clebsch-Gordan coefficients $C^{\\alpha ab}=C^{\\alpha ba}$ and totally symmetric $d^{\\alpha \\beta \\gamma}$ constants, which are only implicitly defined through the two relations\n\n\\begin{align}\n&C^{\\alpha ab}C^{\\alpha cd} = f^{ace}f^{edb}+ f^{ade}f^{ecb}\\,, \\nn \\\\\n&C^{\\alpha ab}d^{\\alpha \\beta \\gamma}= (T^a)^{\\beta \\alpha} (T^b)^{\\alpha \\gamma}+ C^{\\beta ac} C^{\\gamma cb} + (a \\leftrightarrow b)\\,.\n\\label{colorConstr}\n\\end{align}\n\n\n\n\nFrom \\eqn{colorConstr}, and together with the Lie algebra relations that trivially follow from infinitesimal group transformations\n\\begin{align}\n&(T^{a})^{\\alpha \\gamma}(T^{b})^{\\gamma \\beta}-(T^{b})^{\\alpha \\gamma}(T^{a})^{\\gamma \\beta}= i f^{abc} (T^{c})^{\\alpha \\beta}\\,,   \\label{1stID} \\\\\n&f^{bae}C^{\\alpha ec}+f^{cae}C^{\\alpha be}=i(T^{a})^{\\alpha \\beta}C^{\\beta bc}\\,, \\label{2ndID}  \\\\\n&(T^{a})^{\\alpha \\delta}d^{\\delta \\beta \\gamma}+(T^{a})^{\\beta \\delta}d^{\\alpha \\delta \\gamma}+(T^{a})^{\\gamma \\delta}d^{\\alpha \\beta \\delta}=0  \\label{3rdID} \\,,\n\\end{align}\n\n\n\\noindent we have a sufficient number of relations to reduce any tree-level Feynman diagram with external adjoint particles (and possibly internal scalars) to a sum over strings of $f^{abc}$ structure constants, or equivalently, a sum over single-trace factors ${\\rm Tr}(T^{a_1} \\cdots T^{a_n})$. So the gluonic amplitudes for this theory can also be expressed as in \\eqref{c-o amp}. Furthermore, the color-ordered amplitudes will obey the Kleiss-Kuijf relations by virtue of the fact that the trees can alternatively be expressed in terms of only $f^{abc}$'s. Hence it is also possible to express the amplitudes of the $(DF)^2$ theory in the DDM basis as well.\n\n\n\nOf course there are significant differences between Yang-Mills and the $(DF)^2$ theory. For instance, the $(DF)^2$ theory will have $1\/p^4$ poles since the kinetic term has four derivatives, and in four dimensions the all-plus and single-minus amplitudes are non-vanishing $A(\\pm + + \\ldots+)\\neq0$. The latter implies that the theory does not admit a supersymmetric generalization, which can also be seen from the presence of the $F^3$ term in the Lagrangian; this operator is well-known to be incompatible with supersymmetry.\n\nBesides the gluon and scalar states, the $(DF)^2$ contain gluon ghost states (i.e. the linearized equations of motion for $A^\\mu$ has additional solutions) which have the wrong-sign propagator. According to standard field-theory arguments this suggest that the $(DF)^2$ theory violates unitarity; however, this will not be important in the current context. As formal objects the tree amplitudes are well defined, and it is not surprising that such ghost states are present given the close relationship between $(DF)^2$ and conformal gravity. The only caveat is that we need to be careful with how the gluon amplitudes are defined. The external gluon states are taken on the usual plane-wave form $\\varepsilon^\\mu e^{i p \\cdot x}$, and for the LSZ prescription we are amputating the Feynman diagrams by isolating the residue of the $1\/p^4$ poles of the external legs.\n\nSome examples of four-gluon amplitudes in $D=4$ are\n\\begin{align}\nA^{(DF)^2}(1^-,2^-,3^+,4^+)&=2i u \\frac{\\spa{1}.{2}^2}{\\spa{3}.{4}^2}\\,,\\nn \\\\\nA^{(DF)^2}(1^-,2^+,3^-,4^+)&=2i  u \\frac{\\spa{1}.{3}^2}{\\spa{2}.{4}^2}\\,, \\nn \\\\\nA^{(DF)^2}(1^+,2^+,3^+,4^+)&=2i u \\frac{\\spb{1}.{2} \\spb{3}.{4}}{\\spa{1}.{2}\\spa{3}.{4}}\\,,\\nn \\\\\nA^{(DF)^2}(1^-,2^+,3^+,4^+)&=2i  \\spb{2}.{4}^2  \\frac{\\spa{1}.{2} \\spb{2}.{3}}{\\spb{1}.{2}\\spa{2}.{3}}\\,.\n\\end{align}\n\n\nNotice how some of the color-ordered amplitudes have $1\/p^4$ poles and one of them has a $u$ pole which is not possible in Yang-Mills for this particular ordering. These amplitudes however still satisfy the BCJ amplitudes relations \\eqref{BCJ amp} and it is possible to write the amplitudes in such a form that they satisfy color-kinematics duality (the relations \\eqref{colorConstr} are necessary for the theory to satisfy the duality, and demanding that the theory satisfy the duality was part of how the color relations were found in \\cite{conformal}). Notice that the denominators, $D_i$, in \\eqref{amp for c-k} will still be the same as they were in Yang-Mills theory, even though this theory contains double propagators. The extra poles will simply be absorbed into the numerator factors.\n\nAs shown in \\cite{conformal} it is possible to get conformal gravity by using the double copy between the $(DF)^2$ and ordinary Yang-Mills. Schematically, we write this as\n\\begin{align}\n{\\rm CG} &= (DF)^2 \\otimes {\\rm YM}\\,.\n\\end{align}\nFor the supersymmetric generalizations (${\\cal N}=1,2,4$ in $D=4$ notation) we get conformal supergravity from the double copy \n\\begin{align}\n{\\rm CSG} &= (DF)^2 \\otimes {\\rm SYM}\\,,\n\\end{align}\nwhere all the supersymmetry belongs to the SYM theory. At tree level and for adjoint external particles, we can write the double copy in terms of the KLT formula,\n\\begin{align}\n{\\cal M}_n^{\\rm C(S)G}&= A^{(DF)^2} \\cdot S \\cdot A^{\\rm (S)YM}\\,.\\label{KLT CSG}\n\\end{align}\n\nAs an example consider the following four-point MHV amplitude in conformal gravity\n\\begin{align}\nM^{\\rm CG}(1^{--},2^{--},3^{++},4^{++}) &=A^{(DF)^2}(1^-,2^-,3^+,4^+) \\Big( -i \\frac{st}{u}\\Big) A^{\\rm YM}(1^-,2^-,3^+,4^+)= i  \\frac{\\spa{1}.{2}^4 \\spb{3}.{4}^4}{s^2}\\,.\n\\end{align}\n\n\nOne can of course do the double copy where both numerators come from the $(DF)^2$ theory. As will hopefully become clear in section \\ref{ambitwist sec}, the resulting theory will be the $(\\rm Weyl)^3$ theory that arises from the bosonic ambitwistor string \\cite{Mason:2013sva}.\n\n\n\n\n\n\n\n\n\n\n\n\\section{The Scattering Equations and the CHY Formula}\\label{scat eq sec}\n\nIt is our goal to express the amplitudes of the theory described in section \\ref{DFtheory} in the CHY formulation. In this section we will therefore review some basics about the CHY formulation as well as some functions that will prove useful when considering the $( DF)^2$ theory.\n\nThe amplitudes of several quite different theories can be written in the following form in $D$ dimensions:\n\n\\begin{align}\n\\mathcal{A}_n=&{}ig^{n-2}\\int \\frac{d^n\\sigma}{\\mathrm{vol[SL}(2,\\mathbb{C})]}\\prod_i\\phantom{}'\\delta\\left(\\sum_{j\\neq i}\\frac{p_i\\cdot p_j}{\\sigma_{ij}}\\right\nI_LI_R\\label{CHY generel}\n\\end{align}\n\nHere the prime on the product sign means that three of the delta function are left out:\n\n\\begin{align}\n\\prod_i\\phantom{}'\\delta\\left(\\sum_{j\\neq i}\\frac{p_i\\cdot p_j}{\\sigma_{ij}}\\right)\\equiv&{}\\sigma_{kl}\\sigma_{lm}\\sigma_{mk}\\prod_{i\\neq k,l,m}\\delta\\left(\\sum_{j\\neq i}\\frac{p_i\\cdot p_j}{\\sigma_{ij}}\\right).\n\\end{align}\n\nThis is necessary as the scattering equations are $\\mathrm{SL}(2,\\mathbb{C})$ invariant. The three factors of $\\sigma$ in the above expression ensures invariance under permutations. Similarly the factor of $\\mathrm{vol[SL}(2,\\mathbb{C})]$ in the denominator is also necessary in order not to integrate over infinitely many identical terms. It indicates that three of the integration variables will have to be fixed. The remaining part of the integrand is divided into two parts: a left integrand and a right integrand. When we turn towards the ambitwistor string theories, these two parts of the integrand will correspond to different parts of the string action.\n\nIn order to get Yang-Mills amplitudes, one can make the following choices for the left and right integrand:\n\n\\begin{align}\nI_L=&{}\\sum_{\\beta\\in S_n\/Z_n}\\frac{\\mathrm{Tr}\\left(T^{a_{\\beta(1)}}T^{a_{\\beta(2)}}\\cdots T^{a_{\\beta(n)}}\\right)}{\\sigma_{\\beta(1)\\beta(2)}\\sigma_{\\beta(2)\\beta(3)}\\cdots \\sigma_{\\beta(n)\\beta(1)}},&I_R=&{}\\mathrm{Pf}'M_n.\\label{YM CHY}\n\\end{align}\n\n\nThe dependence on the polarization vectors in the amplitude comes from the $2n\\times2n$ antisymmetric matrix called $M_n$. This matrix can be written in the following form:\n\n\\begin{align}\nM_n=&{}\\left(\\begin{array}{cc}\nM_A&-M_C^T\\\\\nM_C&M_B\n\\end{array}\n\\right),\\label{MatrixM_n}\n\\end{align}\n\\noindent where the different submatrices are defined as:\n\\begin{align}\nM_{A,n}^{i,j}&=\\left\\{\\begin{array}{cc}\\frac{p_i\\cdot p_j}{\\sigma_{ij}}&\\text{for }i\\neq j\\\\\n0&\\text{for }i=j\\end{array}\\right.,&M_{B,n}^{i,j}&=\\left\\{\\begin{array}{cc}\\frac{\\varepsilon_i\\cdot \\varepsilon_j}{\\sigma_{ij}}&\\text{for }i\\neq j\\\\\n0&\\text{for }i=j\\end{array}\\right.\\label{SubMatricesM_n}\\\\\nM_{C,n}^{i,j}&=\\left\\{\\begin{array}{cc}\\frac{\\varepsilon_i\\cdot p_j}{\\sigma_{ij}}&\\text{for }i\\neq j\\\\\n-\\sum_{k\\neq i}\\frac{\\varepsilon_i\\cdot p_k}{\\sigma_{ik}}&\\text{for }i=j\\end{array}\\right..\\nonumber\n\\end{align}\n\nThe Pfaffian of this matrix vanishes so the object appearing in the CHY formula is the reduced Pfaffian which is defined by removing rows and columns number $k$ and $l$, then computing the Pfaffian of this smaller matrix and finally multiplying by $(-1)^{k+l}\/\\sigma_{kl}$. The choice of $k$ and $l$ is arbitrary.\n\nIf one instead is interested in the amplitudes of Einstein gravity, one can choose both the left and the right integrand to be given by reduced Pfaffians:\n\n\\begin{align}\nI_L=&{}\\mathrm{Pf}'M_n,&I_R=&{}\\mathrm{Pf}'M_n.\\label{Einstein Gravity}\n\\end{align}\n\nIf on the other hand, one chooses both the left and the right integrand to be given by a color trace over a Parke-Taylor factor:\n\n\\begin{align}\nI_L=&{}\\sum_{\\beta\\in S_n\/Z_n}\\frac{\\mathrm{Tr}\\left(T^{a_{\\beta(1)}}T^{a_{\\beta(2)}}\\cdots T^{a_{\\beta(n)}}\\right)}{\\sigma_{\\beta(1)\\beta(2)}\\sigma_{\\beta(2)\\beta(3)}\\cdots \\sigma_{\\beta(n)\\beta(1)}},&I_R=&{}\\sum_{\\beta\\in S_n\/Z_n}\\frac{\\mathrm{Tr}\\left(T^{a_{\\beta(1)}}T^{a_{\\beta(2)}}\\cdots T^{a_{\\beta(n)}}\\right)}{\\sigma_{\\beta(1)\\beta(2)}\\sigma_{\\beta(2)\\beta(3)}\\cdots \\sigma_{\\beta(n)\\beta(1)}},\n\\end{align}\n\\noindent one will end up with the amplitudes of a bi-adjoint scalar.\n\n\\subsection{Some useful building blocks}\n\nIn order to write the amplitudes for the $(DF)^2$ theory from section \\ref{DFtheory} in the CHY form, it is necessary to use some additional building blocks, besides the ones that Yang-Mills and gravity amplitudes are constructed from. These building blocks must contain an additional factor of momentum squared as compared to the reduced Pfaffian used for Yang-Mills amplitudes. This can easily be seen by inspecting the Lagrangian: the term with three gluons also contains three derivatives (as opposed to one for Yang Mills), the term with four gluons contains two derivatives (as opposed to none for Yang Mills) etc. Fortunately such factors have already been discussed in the literature \\cite{Lam:2016tlk,He:2016iqi}. They can be written in terms of the following functions:\n\n\\begin{align}\nw_{(i_1i_2\\cdots i_k)}=&{}\\frac{\\frac{1}{2}\\mathrm{tr}\\left(f_{i_1}f_{i_2}\\cdots f_{i_k}\\right)}{\\sigma_{i_1i_2}\\sigma_{i_2i_3}\\cdots \\sigma_{i_ki_1}},\\label{Psi more indices}\n\\end{align}\nwhere the trace is over Lorentz indices and the $f$'s are linearized field strengths:\n\n\\begin{align}\nf_{i}^{\\mu\\nu}=&{}p_{i}^\\mu\\varepsilon_{i}^\\nu-p_{i}^\\nu\\varepsilon_{i}^\\mu.\n\\end{align}\n\nOne also needs to introduce the following special case:\n\n\\begin{align}\nw_{(i)}=&{}-\\sum_{j\\neq i}\\frac{\\varepsilon_i\\cdot p_j}{\\sigma_{ij}}\\label{Psi one index}.\n\\end{align}\n\nA useful feature of these functions is that they are gauge-invariant. Equation \\eqref{Psi more indices} is manifestly gauge-invariant because the linearized field strengths are while equation \\eqref{Psi one index} is gauge-invariant on the support of the scattering equations. It can however be a good idea to rewrite \\eqref{Psi one index} in order to make M\\\"obius invariance manifest in the formula for the amplitude. Therefore we employ momentum conservation to re-express the function as:\n\n\\begin{align}\nw_{(i)}=&{}\\sum_{j\\neq i}\\frac{\\varepsilon_i\\cdot p_j\\sigma_{jr}}{\\sigma_{ri}\\sigma_{ij}}.\n\\end{align}\n\nHere $r$ is simply some external leg which is different from $i$. This is a better way of writing the function because $\\sigma_i$ then appears twice in the denominator just like in the other functions in \\eqref{Psi more indices}, making it easier to construct manifestly M\\\"obius invariant quantities.\n\nFrom the above elements we construct the following permutationally invariant functions to be used when constructing the $n$-pt. amplitudes:\n\n\\begin{align}\nW_{i_1i_2\\cdots i_k}=&{}\\sum_{\\beta\\in S_n}\\frac{1}{i_1i_2\\cdots i_r}w_{(\\beta(1)\\cdots \\beta(i_1))}w_{(\\beta(i_1+1)\\cdots \\beta(i_1+i_2))}\\cdots w_{(\\beta(i_1+i_2\\cdots+i_{k-1}+1)\\cdots \\beta(n))}\\label{P functions}\n\\end{align}\n\nHere the $i$'s are chosen to satisfy:\n\n\\begin{align}\n&i_1\\leq i_2\\cdots\\leq i_k,&i_1+i_2\\cdots+i_k&=n.\n\\end{align}\n\nThese functions have exactly the right number of momenta: $W_{111}$, $W_{12}$ and $W_3$ all contain three momentum vectors while $W_{1111}$, $W_{112}$, $W_{22}$, $W_{13}$ and $W_4$ all contain four momentum vectors. This exactly matches the counting mentioned above. We thus expect that the right integrand will consist of these functions in place of the reduced Pfaffian while the left integrand will remain the color trace over a Parke-Taylor factor just like in Yang-Mills. Indeed this expectation will turn out to be correct.\n\nOne should notice that the functions defined in equation \\eqref{P functions} are not all independent. They can be combined to give the Pfaffian of the matrix $M_n$ which as mentioned before is zero:\n\n\\begin{align}\nM_n=&{}\\sum_{1\\leq i_1\\leq i_2\\cdots \\leq i_k\\leq n}(-1)^{n-k}W_{i_1i_2\\cdots i_k}.\\label{vanishing Pfaffian}\n\\end{align}\n\nAs a consequence of this one gets that:\n\n\\begin{align}\n0=&{}W_{111}-W_{12}+W_3,\\nonumber\\\\\n0=&{}W_{1111}-W_{112}+W_{13}+W_{22}-W_4,\\label{relations among Ws}\\\\\n0=&{}W_{11111}-W_{1112}+W_{113}+W_{122}-W_{14}-W_{23}+W_5,\\nonumber\\\\\n0=&{}W_{111111}-W_{11112}+W_{1113}+W_{1122}-W_{114}-W_{123}-W_{222}+W_{15}+W_{24}+W_{33}-W_6.\\nonumber\n\\end{align}\n\nBecause of these relations there can be different ways of expressing the amplitudes. We will try to write the amplitudes in a way that makes the generalization to $n$-point amplitudes as straigthforward as possible.\n\n\\section{The Amplitudes}\\label{amp sec}\n\nHaving described the CHY formalism as well as some functions that will prove useful, we can now turn our attention to the amplitudes of the $(DF)^2$ theory described in section \\ref{DFtheory}. We have computed the amplitudes up to 6 points using standard Feynman rules and then subsequently determined which of the previously described functions matched them. The expressions in the CHY formalism were evaluated using the tools developed in \\cite{Baadsgaard:2015voa,Bjerrum-Bohr:2016juj} (how to apply these tools to double poles has also been dealt with in \\cite{Zhou:2017mfj}). We arrive at the following results for the amplitudes:\n\n\\begin{align}\n\\mathcal{A}_3^{(DF)^2}=&{}{-}4ig\\int\\!\\!\\! \\frac{d^3\\sigma}{\\mathrm{vol[SL}(2,\\mathbb{C})]}\\prod_i\\phantom{}'\\delta\\left(\\sum_{j\\neq i}\\frac{p_i\\cdot p_j}{\\sigma_{ij}}\\right)\\sum_{\\beta\\in S_3\/Z_3}\\frac{\\mathrm{Tr}\\left(T^{a_{\\beta(1)}}T^{a_{\\beta(2)}}T^{a_{\\beta(3)}}\\right)}{\\sigma_{\\beta(1)\\beta(2)}\\sigma_{\\beta(2)\\beta(3)}\\sigma_{\\beta(3)\\beta(1)}}W_{111},\\label{Amp3}\\\\\n\\mathcal{A}_4^{(DF)^2}=&{}{-}4ig^2\\int\\!\\!\\! \\frac{d^4\\sigma}{\\mathrm{vol[SL}(2,\\mathbb{C})]}\\prod_i\\phantom{}'\\delta\\left(\\sum_{j\\neq i}\\frac{p_i\\cdot p_j}{\\sigma_{ij}}\\right)\\sum_{\\beta\\in S_4\/Z_4}\\frac{\\mathrm{Tr}\\left(T^{a_{\\beta(1)}}T^{a_{\\beta(2)}}\\cdots T^{a_{\\beta(4)}}\\right)}{\\sigma_{\\beta(1)\\beta(2)}\\sigma_{\\beta(2)\\beta(3)}\\cdots \\sigma_{\\beta(4)\\beta(1)}}W_{1111},\\\\\n\\mathcal{A}_5^{(DF)^2}=&{}{-}4ig^3\\int\\!\\!\\! \\frac{d^5\\sigma}{\\mathrm{vol[SL}(2,\\mathbb{C})]}\\prod_i\\phantom{}'\\delta\\left(\\sum_{j\\neq i}\\frac{p_i\\cdot p_j}{\\sigma_{ij}}\\right)\\sum_{\\beta\\in S_5\/Z_5}\\frac{\\mathrm{Tr}\\left(T^{a_{\\beta(1)}}T^{a_{\\beta(2)}}\\cdots T^{a_{\\beta(5)}}\\right)}{\\sigma_{\\beta(1)\\beta(2)}\\sigma_{\\beta(2)\\beta(3)}\\cdots \\sigma_{\\beta(5)\\beta(1)}}W_{11111},\\\\\n\\mathcal{A}_6^{(DF)^2}=&{}{-}4ig^4\\int\\!\\!\\! \\frac{d^6\\sigma}{\\mathrm{vol[SL}(2,\\mathbb{C})]}\\prod_i\\phantom{}'\\delta\\left(\\sum_{j\\neq i}\\frac{p_i\\cdot p_j}{\\sigma_{ij}}\\right)\\sum_{\\beta\\in S_6\/Z_6}\\frac{\\mathrm{Tr}\\left(T^{a_{\\beta(1)}}T^{a_{\\beta(2)}}\\cdots T^{a_{\\beta(6)}}\\right)}{\\sigma_{\\beta(1)\\beta(2)}\\sigma_{\\beta(2)\\beta(3)}\\cdots \\sigma_{\\beta(6)\\beta(1)}}W_{111111}.\n\\end{align}\n\n\nAs mentioned, equations \\eqref{relations among Ws} allow us to write the amplitudes in different ways. At 3 points there is furthermore the additional property that all products of momenta are zero because of the special 3-point kinematics. This means that the functions at 3-point become proportional to each other, $W_3\\propto W_{12}\\propto W_{111}$. There are therefore several ways of representing the amplitudes. The reason for the choices above is that they expose a rather simple pattern which is easy to generalize to $n$ points.\n\nBased on the amplitudes above, we propose the following expression for the $n$-point amplitude:\n\n\n\n\\begin{align}\n\\mathcal{A}_n^{ (DF)^2}\\!\\!=&{}{-}4ig^{n-2}\\int \\frac{d^n\\sigma}{\\mathrm{vol[SL}(2,\\mathbb{C})]}\\prod_i\\phantom{}'\\delta\\left(\\sum_{j\\neq i}\\frac{p_i\\cdot p_j}{\\sigma_{ij}}\\right)\\!\\!\\sum_{\\beta\\in S_n\/\\sigma_n}\\!\\!\\frac{\\mathrm{Tr}\\left(T^{a_{\\beta(1)}}T^{a_{\\beta(2)}}\\cdots T^{a_{\\beta(n)}}\\right)}{\\sigma_{\\beta(1)\\beta(2)}\\sigma_{\\beta(2)\\beta(3)}\\cdots \\sigma_{\\beta(n)\\beta(1)}}W_{\\underbrace{11\\cdots1}_n}\\label{DF^2 CHY},\n\\end{align}\n\nThis equation is somewhat similar to the formula for the Yang-Mills amplitudes, only with the reduced Pfaffian, $\\mathrm{Pf}'M_n$, replaced by the function $4W_{11\\cdots1}$. A curious property of this formula is that it contains no $\\varepsilon_i\\cdot\\varepsilon_j$-terms. This has interesting consequences upon dimensional reduction. Consider the case where we go from $D$ dimensions to $d$ dimensions. The $D$-dimensional gluon then splits into a $d$-dimensional gluon and $D-d$ scalars. However the lack of any $\\varepsilon_i\\cdot\\varepsilon_j$-terms in the amplitudes tells us that the new scalars decouple. This property is not manifest in the Feynman rules and only appears after many different terms cancel each other.\n\n\nIn order to support the claim that \\eqref{DF^2 CHY} is in fact the correct $n$-point generalization, we are going to check that it has the correct factorization channels. Since we only have a formula for the scattering of $n$ gluon fields and none with the scalars in the theory as external states, we are going to focus on how the amplitude factorizes when a gluon goes on-shell. These are in any case the easiest factorization channels to determine since they provide a double pole when $q^2\\to0$ as opposed to the scalars which only give a single pole.\n\n\n\n\n\n\n\\subsection{Factorization}\\label{fact sub}\n\nIn order to check the factorization channels of \\eqref{DF^2 CHY}, the external momenta are divived into two groups:\n\n\\begin{align}\nL&=\\{1,\\cdots, n_L\\},&R&=\\{n_L+1,\\cdots, n\\}.\n\\end{align}\n\nWe then consider the case where the sum of the momenta in each group goes on-shell:\n\n\\begin{align}\n\\sum_{i=1}^{n_L}p_i&\\equiv q_R,\\\\\n\\sum_{i=n_L+1}^np_i&\\equiv q_L=-q_R,\\\\\nq_R^2&\\to0.\n\\end{align}\n\nIf \\eqref{DF^2 CHY} is the correct $n$-point generalization, the formula should develop a $q_R^{-4}$-pole and the residue of this pole be the product of two lower-point amplitudes of the same form.\nThis will turn out to indeed be the case as can be demonstrated by considering different pieces of the formula individually.\n\nAs shown in \\cite{Cachazo:2013iea}, the trick to study a factorization channel like the one above is to redefine the integration variables:\n\n\\begin{align}\n\\sigma_i&=\\frac{s}{u_i},&\\text{for }i&\\in L,\\\\\n\\sigma_i&=\\frac{v_i}{s},&\\text{for }i&\\in R.\n\\end{align}\n\nThe variables $u_1$, $u_2$ and $v_n$ will be fixed in order to remove the $\\mathrm{SL}(2,\\mathbb{C})$ symmetry from the amplitude expression. In addition to this, the variable $v_{n-1}$ will be consider to be fixed in exchange for treating $s$ as an integration variable. This means that now four $u$, $v$ variables are fixed. However one would expect there to be six (three for each amplitude). The last two of the fixed integration variables will be the ones corresponding to the new states arising from letting $q_R^2$ go on-shell, and in the calculations to come the quantities will factorize into pieces that will look exactly as expected if the $u$ and $v$ variables corresponding to the new on-shell states have been set to zero.\n\nThe $s$ integration will be responsible for the pole. When $q_R^2$ goes to zero, the variable will begin to behave like\n\n\\begin{align}\ns^2&\\sim\\frac{q_R^2}{\\sum_{i\\in R}\\frac{v_n-v_i}{v_n}\\sum_{j\\in L}\\frac{2p_i\\cdot p_j}{u_jv_i}}\\label{s^2 behave}.\n\\end{align}\n\nThe order of the pole (or whether there is one) then depends on how many factors of $s$ come from the different parts of the CHY expression. The individual factors will be dealt with in appendix \\ref{fact app}. We will only be interested in the dominant terms which will be the ones with the lowest power of $s$. Below is a summary of the powers of $s$ for the $(DF)^2$ theory contrasted with ordinary Yang-Mills:\n\n\n\n\\begin{center}\n\n\\begin{tabular}{c|c|c}\n&$(DF)^2$ &  Yang-Mills  \\label{side med tabel}   \\\\\n\\hline \n \\hline \n   $\\frac{d^n\\sigma}{\\mathrm{vol[SL}(2,\\mathbb{C})]}$                      & $s^{n_L-n_R-3}$   &  $s^{n_L-n_R-3}$\\\\ \\hline \n $\\prod_i\\phantom{}'\\delta\\left(\\sum_{j\\neq i}\\frac{p_i\\cdot p_j}{\\sigma_{ij}}\\right)$               &    $s^{n_L-n_R-2}$   &    $s^{n_L-n_R-2}$ \\\\  \\hline \n $\\frac{\\mathrm{Tr}\\left(T^{a_{1}}T^{a_{2}}\\cdots T^{a_{n}}\\right)}{\\sigma_{12}\\sigma_{23}\\cdots \\sigma_{n1}}$                &          $s^{-n_L+n_R+2}$        &  $s^{-n_L+n_R+2}$\\\\ \\hline\n$\\mathrm{Pf}'M_n$ & --- & $s^{-n_L+n_R+2}$ \\\\ \\hline\n$W_{\\underbrace{11\\cdots1}_n}$ & $s^{-n_L+n_R}$ & --- \\\\ \n\\hline \n \\hline \nTotal & $s^{-3}$ & $s^{-1}$\n\\end{tabular}\n\n\\end{center}\n\n\nWe see that the $(DF)^2$ theory has an extra factor of $s^{-2}$ compared to Yang-Mills, which is to be expected since this theory has double poles while Yang-Mills only has single poles. In the $q_R^2\\to0$ limit, the amplitude of the $(DF)^2$ theory then become proportional to\n\n\\begin{align}\n\\int ds \\frac{1}{s^3}\\delta\\left(\\sum_{i\\in R}\\frac{v_n-v_i}{v_n}\\sum_{j\\in L}\\frac{2p_i\\cdot p_j}{u_jv_i}s^2-q_R^2\\right)=&{}\\frac{\\sum_{i\\in R}\\frac{v_n-v_i}{v_n}\\sum_{j\\in L}\\frac{p_i\\cdot p_j}{u_jv_i}}{q_R^4}.\\label{ds}\n\\end{align}\n\nThe numerator can be understood as the product of the $w_{(i)}$-functions for the new on-shell state. We therefore introduce polarization vectors for the intermediate state that has gone on-shell:\n\n\\begin{align}\n\\epsilon_{q_L}\\cdot q_L&=0,\\\\\n\\epsilon_{q_R}\\cdot q_R&=0,\\\\\n\\sum_{+\/-}\\epsilon_{q_L}^\\mu\\epsilon_{q_R}^\\nu&=-2\\eta^{\\mu\\nu}+\\cdots.\n\\end{align}\n\nHere $\\cdots$ indicate terms proportional to $q_L^\\mu$ or $q_R^\\nu$. These terms vanish as each lower point amplitude is gauge-invariant. Equation \\eqref{ds} can then be written as:\n\n\\begin{align}\n\\int ds\\frac{1}{s^3}\\delta\\left(\\sum_{i\\in R}\\frac{v_n-v_i}{v_n}\\sum_{j\\in L}\\frac{2p_i\\cdot p_j}{u_jv_i}s^2-q_R^2\\right)=&{}\\sum_{+\/-}\\frac{\\sum_{i\\in R}\\frac{\\epsilon_{q_R}\\cdot p_i(v_n-v_i)}{v_iv_n}\\sum_{j\\in L}\\frac{\\epsilon_{q_L}\\cdot p_j}{u_j}}{q_R^4}\\label{ds2}\n\\end{align}\n\nThe numerator is equivalent to two $w_{(i)}$-functions with the $u$ and $v$ variables corresponding to the new on-shell state both having been fixed to 0.\n\nThe remaining details can be found in appendix \\ref{fact app}. Putting them all together, one arrives at the conclusion that \\eqref{DF^2 CHY} does indeed satisfy the correct factorization properties:\n\n\\begin{align}\n\\mathcal{A}_n^{ (DF)^2}(L,R)\\bigg|_{q_R^2\\to0}=&{}\\sum_{+\/-}\\mathcal{A}_{n_L}^{ (DF)^2}(L,q_L^{a_L})\\frac{-i\\delta^{a_{q_L}a_{q_R}}}{q_R^4}\\mathcal{A}_{n-n_L}^{ (DF)^2}(q_R^{a_R},R)\\label{amp fact}\n\\end{align}\n\nAs a final comment about factorization, let us focus on some terms that do not play a role in \\eqref{amp fact}, but are nonetheless interesting. They are some of the sub-leading terms from the color part of the CHY formula. The only terms that contribute to \\eqref{amp fact} are those where the color generators in the trace separate nicely into one product of generators for the $L$ set and one product of generators for the $R$ set. As a shorthand, we could denote these as the $\\mathrm{Tr}(LR)$-terms. However, one could also consider the $\\mathrm{Tr}(LRLR)$-terms. Such terms do not generate a pole in Yang-Mills theory as they correspond to having an intermediate state which is not in the adjoint representation of the gauge group. However they do generate a simple pole in the $(DF)^2$ theory, which is to be expected since this theory does in fact contain particles that are not in the adjoint representation, the scalars.\n\nTo conclude, this section showed that \\eqref{DF^2 CHY} factorizes into two amplitudes of the same form when a $1\/q^4$ propagator was put on-shell. It also showed that the expression for the amplitude requires that the theory contain particles that are in a different representation of the gauge group than the adjoint. Both these observations support the claim that \\eqref{DF^2 CHY} is in fact the correct $n$-point amplitude for the $(DF)^2$ theory.\n\n\n\n\n\\subsection{Conformal gravity amplitudes}\n\nConformal gravity can be found through combining the $(DF)^2$ theory described in section \\ref{DFtheory} with standard super Yang-Mills in the KLT relations \\cite{Kawai:1985xq}. In the CHY formalism, one can simply replace the color factor in \\eqref{DF^2 CHY} with the reduced Pfaffian from Yang-Mills:\n\n\\begin{align}\n\\mathcal{A}_n^{\\rm CG}=&{}\\int \\frac{d^n\\sigma}{\\mathrm{vol[SL}(2,\\mathbb{C})]}\\prod_i\\phantom{}'\\delta\\left(\\sum_{j\\neq i}\\frac{p_i\\cdot p_j}{\\sigma_{ij}}\\right)W_{\\underbrace{11\\cdots1}_n}\\mathrm{Pf}'M_n\\label{CHY CG}\n\\end{align}\n\nThis should be the $D$-dimensional formula for conformal gravity (up to some overall constant). As a simple check for this formula let us point out that using the factorization properties of the reduced Pfaffian:\n\n\\begin{align}\n\\mathrm{Pf}'M_n\\sim s^{n_R-n_L+2}\\left(\\prod_{i=1}^{n_L}u_i^2\\right)\\mathrm{Pf}'M_L\\mathrm{Pf}'M_R,\n\\end{align}\n\\noindent it is straightforward to show that the formula factorizes correctly\n\nAnother simple check is to focus on the 4-dimensional MHV amplitudes. It is believed that in this case, there is only one relevant solution to the scattering equations\t\\cite{Monteiro:2013rya,Weinzierl:2014vwa,Naculich:2014naa}. It can be written in terms of spinors as follows:\n\n\\begin{align}\n\\sigma_i=&{}\\frac{\\langle i1\\rangle\\langle2\\chi\\rangle}{\\langle i\\chi\\rangle\\langle21\\rangle}.\n\\end{align}\n\nHere $|\\chi\\rangle$ is an arbitrary spinor not collinear with $|1\\rangle$ or $|2\\rangle$. This solution was proven to give the correct $n$-point MHV amplitude for Yang-Mills theory and Einstein gravity in \\cite{Du:2016blz} where it was also shown that, at least up to 9-point, the other solutions to the scattering equations make the reduced Pfaffian vanish. Compared to those two theories, the only new element in \\eqref{CHY CG} is the function $W_{11\\cdots1}$ which, on this particular solution to the scattering equations and assuming that particles 1 and 2 are the only negative helicity gluons, can be written as:\n\n\\begin{align}\nW_{\\underbrace{11\\cdots1}_n}=&{}\\left(\\frac{\\langle21\\rangle}{\\langle1\\chi\\rangle\\langle2\\chi\\rangle}\\right)^n\\left(\\prod_{i=1}^n\\langle i\\chi\\rangle^2\\right)\\left(\\prod_{j=3}^n\\sum_{k\\neq j}\\frac{[jk]\\langle k\\eta\\rangle^2}{\\langle jk\\rangle\\langle j\\eta\\rangle^2}\\right),\\label{W MHV}\n\\end{align}\n\\noindent where $|\\eta\\rangle$ is another arbitrary spinor not necessarily identical to $|\\chi\\rangle$.\n\nBy combining \\eqref{W MHV} with results from the previously mentioned papers, one easily arrives at the following results for the scattering of gravitons among themselves and scattering between gravitons and scalars:\n\n\\begin{align}\n\\mathcal{A}_n^{\\rm CG}(1^-2^-3^+\\cdots n^+)=&{}\\langle12\\rangle^4\\left(\\prod_{j=3}^n\\sum_{k\\neq j}\\frac{[jk]\\langle k\\eta\\rangle^2}{\\langle jk\\rangle\\langle j\\eta\\rangle^2}\\right),\\nonumber\\\\\n\\mathcal{A}_n^{\\rm CG}(1^-2^\\varphi 3^+\\cdots n^+)=&{}\\langle12\\rangle^4\\left(\\prod_{j=2}^n\\sum_{k\\neq j}\\frac{[jk]\\langle k\\eta\\rangle^2}{\\langle jk\\rangle\\langle j\\eta\\rangle^2}\\right),\\\\\n\\mathcal{A}_n^{\\rm CG}(1^\\varphi 2^\\varphi 3^+\\cdots n^+)=&{}\\langle12\\rangle^4\\left(\\prod_{j=1}^n\\sum_{k\\neq j}\\frac{[jk]\\langle k\\eta\\rangle^2}{\\langle jk\\rangle\\langle j\\eta\\rangle^2}\\right)\\nonumber\n\\end{align}\n\nThese amplitudes exactly match the expression found by Berkovits and Witten \\cite{Berkovits:2004jj}, and thus we see that \\eqref{CHY CG} can in fact be seen as a $D$-dimensional generalization of the Berkovits-Witten formula.\n\nAs a side note let us point out that this way of simplifying the CHY formulation in 4-dimensional MHV case will not work for the $(DF)^2$ theory. This is due to the fact that the function $W_{11\\cdots1}$ is just a product of functions for each individual on-shell leg (the $w_{(i)}$'s from equation \\eqref{Psi one index}), which means that for the function to be zero, one of these functions will have to be zero. These functions only depend on the helicity of the given external leg and not on all the other helicities. So if we imagine that a given solution to the scattering equations does not contribute to the all plus amplitudes because it sets $w_1$ to 0, then all other amplitudes where the helicity of particle 1 is positive will also not get contributions from this solution to the scattering equations.\n\nWe should also note that of supersymmetrizing \\eqref{CHY CG} is essentially the same as the problem for Yang-Mills theory since the supersymmetry in the $R^2$ theory derives from this theory (see equation \\eqref{KLT CSG}). If it is possible to construct a simple CHY-formulation for the amplitudes of $\\mathcal{N}=4$ super Yang-Mills, it should therefore be straightforward to construct a supersymmetric version of equation \\eqref{CHY CG} as well.\n\n\\section{Ambitwistor Interpretation}\\label{ambitwist sec}\n\nThe fact that the\n amplitudes of conformal gravity and the $(DF)^2$ theory can be written as CHY formulae suggests that there should be ambitwistor string theories \\cite{Mason:2013sva,Casali:2015vta} corresponding to them.\nIn this section we will briefly review ambitwistor string theory and show which specific choices of the worldsheet action lead to the amplitudes given in equations \\eqref{DF^2 CHY} and \\eqref{CHY CG}.\n\n\\subsection{Review}\n\nThe ambitwistor string theories  can be thought of as chiral worldsheet models describing the interactions of massless states. In the simplest example, bosonic strings, the action is given by\n\\begin{equation}\nS_\\mathrm{B} = \\frac{1}{2\\pi}\\int \\mathrm{d}^2 \\sigma \\left( P_\\mu \\bar\\partial X^\\mu - \\frac{1}{2}eP^2\\right),\n\\label{bosambi}\n\\end{equation}\nwhere $X^\\mu$  ($\\mu = 0$ to $D-1$) denotes the string coordinates in the $D$-dimensional target space, $P_\\mu$ are their conjugate momenta and $e$ is a Lagrange multiplier enforcing the constraint $P^2=0$.\n\nBecause of this first-class constraint, the model is invariant under the following local symmetry, in addition to reparameterization invariance:\n\\begin{equation}\n\\delta X^\\mu = \\alpha P^\\mu\\,, \\qquad \\delta P_\\mu = 0\\,, \\qquad \\delta e = \\bar\\partial \\alpha\\,,\n\\end{equation}\nfor some transformation parameter $\\alpha$. One can use this symmetry to gauge-fix $e = 0$, and then the standard BRST procedure yields the gauge-fixed action\n\\begin{equation}\nS_\\mathrm{B}^\\star = \\frac{1}{2\\pi}\\int \\mathrm{d}^2 \\sigma \\left( P_\\mu \\bar\\partial X^\\mu + b\\bar\\partial c + \\widetilde{b}\\bar\\partial\\widetilde{c}\\right),\n\\end{equation}\ntogether with the BRST charge\n\\begin{equation}\nQ = \\frac{1}{2\\pi\\mathrm{i}}\\oint \\mathrm{d}\\sigma \\left(cT - bc\\partial c +\\frac{1}{2}\\widetilde{c}P^2\\right),\n\\label{BRSTbos}\n\\end{equation}\nwhere $T$ is the complete energy-momentum tensor (matter $+$ ghosts), $(b,c)$ are the usual (anti)ghosts of string theory and $(\\widetilde{b},\\widetilde{c})$ are the (anti)ghosts corresponding to the extra gauge symmetry.\n\nPhysical states correspond to vertex operators in the cohomology of $Q$, which in this case contains only\\footnote{In this paper we consider only plane-wave states, even though higher-derivative theories typically contain other types of states such as those of the form $A\\cdot X\\, \\mathrm{e}^{\\mathrm{i} p\\cdot X}$.}\n\\begin{equation}\nV = c \\widetilde{c}\\,P_\\mu P_\\nu \\epsilon^{\\mu\\nu} \\mathrm{e}^{\\mathrm{i} p\\cdot X}\n\\label{Vbos}\n\\end{equation}\nand its integrated version\n\\begin{equation}\nU = \\int \\mathrm{d}^2 \\sigma\\,\\bar\\delta(p\\cdot P) \\,P_\\mu P_\\nu \\epsilon^{\\mu\\nu} \\mathrm{e}^{\\mathrm{i} p\\cdot X}\\,.\n\\label{Ubos}\n\\end{equation}\nBRST-closedness requires $p^2 = p_\\mu\\epsilon^{\\mu\\nu}=0$, while the analysis of BRST-exact states implies the gauge transformation $\\delta \\epsilon^{\\mu\\nu} = p^{(\\mu}\\epsilon^{\\nu)}$ for some parameter $\\epsilon^\\mu$ such that $p_\\mu\\epsilon^\\mu = 0$. Thus, these operators correspond to an on-shell graviton.\n\n\nHowever, if one computes the correlation function containing three unintegrated vertex operators, the result does not agree with the expected three-point amplitude coming from Einstein gravity. In fact, it is of order six in the momenta. In \\cite{Mason:2013sva}, the authors could not interpret the result in terms of any known theory of gravity, although they mention that it could be related to a (Weyl)${}^3$ vertex. The tree-level $n$-point function is given by\n\\begin{equation}\n\\mathcal{A}_n=\\int \\frac{d^n\\sigma}{\\mathrm{vol[SL}(2,\\mathbb{C})]}{\\prod_i}^\\prime\\bar\\delta\\left( p^{(i)}\\cdot P(\\sigma_i)\\right) \\prod_{j=1}^n \\epsilon_{(j)}^{\\mu\\nu}P_\\mu(\\sigma_j)P_\\nu(\\sigma_j)\\,,\n\\end{equation}\nwith $P_\\mu$ constrained to take its value as $P_\\mu(\\sigma) = \\sum_{i=1}^n p^{(i)}_\\mu\/(\\sigma-\\sigma_i)$. Note that, using the language introduced in section 3, this amplitude can be cast as\n\\begin{equation}\n\\mathcal{A}_n=\\int \\frac{d^n\\sigma}{\\mathrm{vol[SL}(2,\\mathbb{C})]}{\\prod_i}^\\prime\\bar\\delta\\left( p^{(i)}\\cdot P(\\sigma_i)\\right) {\\left(W_{\\underbrace{11\\cdots1}_n}\\right)}^2,\n\\end{equation}\nand the appearance of the $W_{11\\cdots1}$ function squared indicates that this theory will be the result of squaring the $(DF)^2$ theory via the double copy.\n\n This purely bosonic model can be generalized in many different ways. To do so, the standard procedure consists of adding two other terms to the action  (\\ref{bosambi}), $S_\\mathrm{L}$ and $S_\\mathrm{R}$, which ultimately correspond to the left and right integrands in CHY formulae (cf. (\\ref{CHY generel})). \n\n\n\n\nIn perhaps the most successful example, both $S_\\mathrm{L}$ and $S_\\mathrm{R}$ are RNS-like fermion systems, with the important difference that in the ambitwistor case all worldsheet fields are left-moving (holomorphic). The complete action is given by:\n\\begin{equation}\nS_\\mathrm{B} + S_\\mathrm{L} + S_\\mathrm{R} = \\frac{1}{2\\pi}\\int \\mathrm{d}^2 \\sigma \\left( P_\\mu \\bar\\partial X^\\mu - \\frac{1}{2}eP^2  +\\frac{1}{2}\\sum_{r=1,2} (\\Psi_{r\\mu} \\bar\\partial \\Psi^\\mu_r - 2\\chi_r P_\\mu\\Psi_r^\\mu)\\right),\n\\label{actiontypeII}\n\\end{equation}\nwhere $\\Psi_1^\\mu, \\Psi_2^\\mu$ are the worldsheet fermions and $\\chi_1, \\chi_2$ are fermionic Lagrange multipliers for the fermionic constraints $P\\cdot\\Psi_1, P\\cdot\\Psi_2$.\n\nGauge-fixing the Lagrange multipliers to zero via the BRST procedure, one ends up with the usual RNS-like bosonic (anti)ghosts $(\\beta_1,\\gamma_1)$ and $(\\beta_2,\\gamma_2)$, in addition to the same (anti)ghosts as before. The BRST charge is now given by\n\\begin{equation}\nQ_{(\\Psi_1, \\Psi_2)} = \\frac{1}{2\\pi\\mathrm{i}}\\oint \\mathrm{d}\\sigma\\left(cT - bc\\partial c +\\frac{1}{2}\\widetilde{c}P^2 +\\sum_r (\\gamma_r\\Psi_r\\cdot P +\\widetilde{b}\\gamma_r\\gamma_r)\\right)\n\\label{BRSTtypeII}\n\\end{equation}\nand its cohomology contains the vertex operator\n\\begin{equation}\nV^{(-1)}_{(\\Psi_1, \\Psi_2)} = c\\widetilde{c} \\,\\mathrm{e}^{\\mathrm{i}p\\cdot X} \\prod_r \\delta(\\gamma_r) \\Psi_r\\cdot \\epsilon_r\\,,\n\\label{vertextypeII}\n\\end{equation}\ntogether with corresponding  picture-number-zero or integrated versions, where $\\epsilon_1^\\mu, \\epsilon_2^\\nu$ combine to form the graviton, Kalb--Ramond and dilaton polarizations.\nOne can show that the tree-level $n$-point correlation function of these vertex operators gives rise to the CHY formula (\\ref{Einstein Gravity}) when restricted to gravitons.\n\n\n\nAnother possibility for $(S_\\mathrm{L}, S_\\mathrm{R})$ is to replace one of the fermionic systems of the previous model with an action for a generic current algebra, $S_\\mathrm{C}$. Then one can define the currents $J_I$ satisfying the OPE\n\\begin{equation}\nJ_I(\\sigma_1) J_J(\\sigma_2) \\sim \\frac{\\ell}{(\\sigma_1-\\sigma_2)^2}\\delta_{IJ} + \\frac{1}{\\sigma_1-\\sigma_2}f_{IJ}{}^K J_K(\\sigma_2)\\,,\n\\end{equation}\nwhere $\\ell$ is the so-called level of the algebra and $f_{IJ}{}^K$ are the structure constants of the gauge group. The BRST charge of this model has the same form as (\\ref{BRSTtypeII}), with the obvious differences that now the sum over $r$ comprises only one term and the energy-momentum tensor is the one corresponding to the new gauge-fixed action.\n\nThis theory is reminiscent of the usual heterotic string theory, and its spectrum also contains two sectors: the gauge one and the gravity one. However, the latter does not correspond to the usual Neveu--Schwarz sector of heterotic strings, and in particular it contains a 3-form potential whose interpretation was unclear in the original work by Mason and Skinner. \n In the gauge sector, the following vertex operator belongs to the cohomology of $Q$:\n\\begin{equation}\nV^{(-1)}_{(\\Psi, J)} = c\\widetilde{c} \\,\\delta(\\gamma) \\Psi\\cdot \\epsilon\\,J_I T^I  \\mathrm{e}^{\\mathrm{i}p\\cdot X}  \\,,\n\\label{V_Psi,J}\n\\end{equation}\nwhere $T^I$ denotes the generators of the gauge group. BRST invariance imposes $p^2=p\\cdot\\epsilon=0$, and the vertex operator is BRST-trivial if $\\epsilon_\\mu \\propto p_\\mu$. Therefore, it describes an on-shell gluon.\n\nWhen restricted to single-trace contributions, the tree-level $n$-point correlation function involving (\\ref{V_Psi,J}) (and the other versions of this vertex operator, as appropriate) is equal to the CHY formula (\\ref{YM CHY}) for gluons.\n\n\n\n\n\n\n\n\n\n\n\n\n\\subsection{\\boldmath{$(S_\\mathrm{L}, 0)$-models}}\\label{s_l,0 afsnit}\n\n\nFrom the review in the previous subsection, it should be clear that there is a correspondence between the choice of $(S_\\mathrm{L}, S_\\mathrm{R})$, the vertex operators and the correlation functions of a given ambitwistor string. We summarize the results presented so far in the following table.\n\n\\begin{table}[h!]\n\\centering\n\\begin{tabular}{c|c|c}\n$S_\\mathrm{L\/R}$ &  Vertex & $I_\\mathrm{L\/R}$    \\\\\n\\hline \n \\hline \n   $0$                      &  $\\epsilon \\cdot P$   &   $W_{\\underbrace{11\\cdots1}_n}$\\\\ \\hline \n $\\Psi$               &    $\\delta(\\gamma) \\epsilon\\cdot\\Psi$   &    Pf${}^\\prime(M_n)$ \\\\  \\hline \n $J$                &          $T^I J_I$        &  color factor\n\\end{tabular}\n\\end{table}\n\n\\noindent In the above, $0$ signifies that $S_\\mathrm{L}$ or $S_\\mathrm{R}$ are absent from the model, e.g. $(0,0)$ represents the bosonic ambitwistor string. Moreover, ``Vertex'' denotes the contribution to the simplest vertex operator and $I_\\mathrm{L\/R}$ the two different parts of the integrand in the CHY formulation of amplitudes (cf. (\\ref{CHY generel})).\n More precisely, the (single-trace) tree-level $n$-point correlation function of any $(S_\\mathrm{L}, S_\\mathrm{R})$-model gives rise to a CHY formula containing $I_\\mathrm{L}$ and $I_\\mathrm{R}$.\n\n\n\nThus, by comparing with \\eqref{DF^2 CHY}, we see that the CHY formula for the $(DF)^2$-theory can be obtained via the ambitwistor model $(J,0)$, while a comparison with \\eqref{CHY CG} leads to the conclusion that the CHY formula for conformal supergravity can be obtained through the  model $(\\Psi,0)$.\n\n\n\n\n\nSince, to the best of our knowledge, models of the type $(S_\\mathrm{L},0)$ have not yet been explored in the literature,  it is worth to discuss them in a bit more detail. In the $(J,0)$ case, the action is given by\n\\begin{equation}\nS_{(J,0)} = \\frac{1}{2\\pi}\\int \\mathrm{d}^2 \\sigma \\left( P_\\mu \\bar\\partial X^\\mu - \\frac{1}{2}eP^2 + \\mathcal{L}_\\mathrm{C}\\right),\n\\label{S_J,0}\n\\end{equation}\nwhere $\\mathcal{L}_\\mathrm{C}$ is the Lagrangian corresponding to a generic current algebra. The gauge-fixing procedure is almost identical to the one for the bosonic case, and we are left with the BRST-charge\n\\begin{equation}\nQ_{(J,0)} = \\frac{1}{2\\pi\\mathrm{i}}\\oint \\mathrm{d}\\sigma\\left(cT - bc\\partial c +\\frac{1}{2}\\widetilde{c}P^2\\right),\n\\label{BRSTJ,0}\n\\end{equation}\nwhich looks exactly the same as (\\ref{BRSTbos}), but now $T$ includes the energy-momentum tensor $T_\\mathrm{C}$ corresponding to $\\mathcal{L}_\\mathrm{C}$. Accordingly, the central charge receives a contribution $c_\\mathrm{C}$ from the gauge sector, and is given by $c_{(J,0)} = 2(D-26) + c_\\mathrm{C}$. Thus, one can  make $c_{(J,0)}$ vanish in a given number of dimensions by choosing the current algebra appropriately. However, we need not concern ourselves much about this since we only work at tree level.\n\n\n\nThe cohomology of $Q_{(J,0)}$ contains the vertex operator\n\\begin{equation}\nV_{(J,0)} = c\\widetilde{c} \\,P \\cdot \\epsilon\\,  \\mathrm{e}^{\\mathrm{i}p\\cdot X}  J_I T^I \\,,\n\\label{V_J,0}\n\\end{equation}\ntogether with its integrated version --- which as usual amounts to replacing the ghosts with $\\int\\mathrm{d}^2 \\sigma\\,\\bar\\delta(p\\cdot P)$. This expression is BRST-invariant if and only if $p^2=p\\cdot\\epsilon=0$, and $\\epsilon^\\mu \\propto p^\\mu$ renders it BRST-trivial, hence it corresponds to an on-shell gluon. It is easy to see that the tree-level $n$-point correlation function computed with these operators gives rise to \\eqref{DF^2 CHY}.\n\n\n\nNote that the cohomology also contains gravity states, a feature common to all known ambitwistor string theories. In this case, the graviton vertex operators are identical to the ones in the bosonic model, given in (\\ref{Vbos}) and (\\ref{Ubos}), and thus the 3-point amplitude exhibits the same (Weyl)${}^3$ behavior. As anticipated in the introduction, it is a general property of $(S_\\mathrm{L},0)$-models that the states and tree-level amplitudes obtainable from one such model can also be obtained from an $(S_\\mathrm{L},J)$-model,\nand the appearance of gravity states in the $(0,J)$-model is just a consequence of that. \nBy the same token, the $(J,0)$-model  can be identified with a sector of the more general $(J,\\tilde{J})$-model, which contains bi-adjoint scalars transforming under two potentially different gauge groups. It is remarkable that the ambitwistor framework allows  such a truncation, i.e. that some sectors can be treated as theories on their own.\nWe will encounter another example of that in the following.\n\n\n\n\n\n\n\n\n\n\n\n\n\n\\vspace{1cm}\n\n\nLet us now discuss the $(\\Psi,0)$ ambitwistor string, which gives rise to the tree-level $n$-point amplitude in \\eqref{CHY CG}.\nThe action of the model is given by\n\\begin{equation}\nS_{(\\Psi,0)} = \\frac{1}{2\\pi}\\int \\mathrm{d}^2 \\sigma \\left( P_\\mu \\bar\\partial X^\\mu - \\frac{1}{2}eP^2  +\\frac{1}{2} \\Psi_{\\mu} \\bar\\partial \\Psi^\\mu - \\chi P_\\mu\\Psi^\\mu\\right).\n\\end{equation}\nAfter gauge-fixing $e=\\chi=0$, one gets the BRST charge\n\\begin{equation}\nQ_{(\\Psi,0)} = \\frac{1}{2\\pi\\mathrm{i}}\\oint \\mathrm{d}\\sigma\\left(cT - bc\\partial c +\\frac{1}{2}\\widetilde{c}P^2 +\\gamma \\Psi^\\mu  P_\\mu +\\widetilde{b}\\gamma^2\\right),\n\\label{BRSTPsi,0}\n\\end{equation}\nwhose cohomology contains the vertex operator\n\\begin{equation}\nV^{(-1)}_{(\\Psi, 0)} = c\\widetilde{c} \\, \\delta(\\gamma)  \\epsilon_1^\\mu \\epsilon_2^\\nu \\Psi_\\mu P_\\nu  \\mathrm{e}^{\\mathrm{i}p\\cdot X}\\,,\n\\end{equation}\ntogether with corresponding  picture-number-zero or integrated versions, where $\\epsilon_1^\\mu, \\epsilon_2^\\nu$ combine to form the graviton, Kalb--Ramond and dilaton polarizations.\nRestricting to gravitons, one can show that\nthe tree-level $n$-point correlation function of these vertex operators gives rise to the CHY formula \\eqref{CHY CG}. However, since the central charge is computed to give $c_{(\\Psi, 0)} = \\frac{5}{2}D-41$, it is not possible to make sense of this model beyond tree level, in any (integer) number of  dimensions.\n\n\n\n\n\nNote that, at tree level, this model is equivalent to the gravity sector of the heterotic ambitwistor string, given by $(\\Psi,J)$. Indeed, the current-algebra part of the heterotic model is inert in the gravity sector, which implies that the cohomology and correlation functions are the same as those in the $(\\Psi,0)$ model. In particular, the $(\\Psi,0)$  model also contains the unexpected (from the Einstein-gravity point of view) massless 3-form first encountered in \\cite{Mason:2013sva}, whose picture-number $-1$ vertex operator is given by\n\\begin{equation}\nV^{(-1)}_{\\mathrm{3-form}}=c\\widetilde{c}\\,\\delta(\\gamma) A_{\\mu\\nu\\rho} \\Psi^\\mu \\Psi^\\nu \\Psi^\\rho \\mathrm{e}^{\\mathrm{i}p\\cdot X}\\,,\n\\end{equation}\nwith $p^\\mu A_{\\mu\\nu\\rho}=0$.\nTherefore, we conclude that the gravity sector of the heterotic ambitwistor string describes conformal supergravity, and  it is then natural to interpret that theory as a generalization of Witten's twistor string theory. We will come back to this point shortly.\n\n\n\n\n\n\n\\vspace{1cm}\n\n\n\n\n\n\n\nFinally, we would like to discuss the more exotic case of the $((\\Psi_1,\\Psi_2),0)$ ambitwistor string. This is reminiscent of the $(\\Psi_1,\\Psi_2)$ model, and indeed the action and BRST operator are the same as (\\ref{actiontypeII}) and (\\ref{BRSTtypeII}), respectively. Hence, one would naively think that the spectrum and correlation functions of the two models are identical.\n\nHowever, putting both fermion systems on the same side of the model translates into having weaker GSO-like conditions. To make this point clearer, consider the following state:\n\\begin{equation}\nV^{(-1)}_{((\\Psi_1,\\Psi_2),\\,0)} = c\\widetilde{c} \\, \\delta(\\gamma_1)  \\delta(\\gamma_2)\\, \\epsilon\\cdot P\\, \\mathrm{e}^{\\mathrm{i}p\\cdot X}\\,.\n\\label{newstate}\n\\end{equation}\nFor $p^2=p\\cdot\\epsilon=0$, this state is BRST-invariant, and $\\epsilon^\\mu \\propto p^\\mu$ renders it trivial, as usual. In the $(\\Psi_1,\\Psi_2)$ model, this state is projected out of the physical spectrum, since in that case one requires physical states to have an even number of $\\{\\gamma_1, \\Psi_1\\}$ and an even number of $\\{\\gamma_2, \\Psi_2\\}$ --- cf. (\\ref{vertextypeII}), for example. In the $((\\Psi_1,\\Psi_2),0)$ model, the GSO-like projection requires that the number of $\\{\\gamma_1,\\gamma_2,\\Psi_1,\\Psi_2\\}$ be even, and thus both (\\ref{vertextypeII}) and (\\ref{newstate})  are considered physical. Since there is no current algebra in this particular model, the state in \\eqref{newstate} corresponds to a U(1)-field, i.e. a photon.\n\nOne can show that the  tree-level $n$-point correlation function of these photon states gives\n\\begin{equation}\n\\mathcal{A}_n=\\int \\frac{d^n\\sigma}{\\mathrm{vol[SL}(2,\\mathbb{C})]}{\\prod_i}^\\prime\\bar\\delta\\left( p^{(i)}\\cdot P(\\sigma_i)\\right) \\left. W_{\\underbrace{11\\cdots1}_n}\\right. {\\left(\\mathrm{Pf}{}^\\prime(M_{A,n})\\right)}^2,\\label{DF^2 photon amplitude}\n\\end{equation}\nwhere $M_{A,n}$ is an $n$ by $n$ matrix identical to one of the submatrices of the bigger matrix $M_n$ defined in \\eqref{SubMatricesM_n}.\nFrom this discussion, it is evident that one more row can be added to the table above \\cite{Casali:2015vta}:\n\\begin{center}\n\\begin{tabular}{c|c|c}\n$S_\\mathrm{L\/R}$ &  Vertex & $I_\\mathrm{L\/R}$    \\\\\n\\hline \n \\hline & \\\\\n   $(\\Psi_1,\\Psi_2)$             &  $\\delta(\\gamma_1)  \\delta(\\gamma_2)$            &  ${\\left(\\mathrm{Pf}{}^\\prime(M_{A,n})\\right)}^2$\n\\end{tabular}\n\\end{center}\n\nLet us now consider the amplitude in \\eqref{DF^2 photon amplitude} from the quantum field theory point of view. It arises from combining the $(DF)^2$ theory with the non-linear sigma model in the KLT relations.\\footnote{The non-linear sigma model corresponds to the $((\\Psi_1,\\Psi_2),J)$  ambitwistor string, as can be seen from the table displayed in the introduction.} By inspecting the amplitude, we find that up to four points the simplest Lagrangian for this theory is given by:\n\n\\begin{align}\n\\tfrac{1}{\\sqrt{-g}}\\mathcal{L}=&{}\\frac{1}{2\\kappa^2}R+\\tfrac{1}{4}\\left(\\nabla_\\mu F_{\\nu\\rho}\\right)\\left(\\nabla^\\mu F^{\\nu\\rho}\\right)+\\tfrac{1}{8}RF_{\\mu\\nu}F^{\\mu\\nu}-\\tfrac{1}{6}\\kappa^2\\left(\\nabla_\\mu F_{\\sigma\\lambda}\\right)\\left(\\nabla^\\sigma F^{\\mu\\nu}\\right)F_{\\nu\\rho}F^{\\rho\\lambda}\\nonumber\\\\\n&+\\tfrac{1}{48}\\kappa^2\\left(\\nabla_\\mu F_{\\nu\\rho}\\right)\\left(\\nabla^\\mu F^{\\nu\\rho}\\right)F_{\\sigma\\lambda}F^{\\sigma\\lambda}+\\cdots.\n\\end{align}\n\nWe will refer to this theory as the $(DF)^2$-photon theory. Note that the ordinary Einstein gravity appears as part of this Lagrangian and that the coupling constant for its self-interaction is the same as for its interaction with the gravitons. From the ambitwistor string theory point of view, the appearance of Einstein gravity is fairly obvious since both the vertices \\eqref{vertextypeII} and \\eqref{newstate} are allowed in the $((\\Psi_1,\\Psi_2),0)$ model. From the quantum field theory perspective, it is less clear how the product of the $(DF)^2$ theory and the non-linear sigma model can give rise to a spin-2 field. Nonetheless, the $(DF)^2$-photon is bound to interact with regular Einstein gravity, as can be seen by the following factorization argument.\n\nConsider an amplitude of $2n$ $(DF)^2$-photons, group the photons into $n$ pairs and take the limit where the propagator for each pair goes on-shell. In this scenario, the amplitude in \\eqref{DF^2 photon amplitude} behaves in the following way:\n\n\\begin{align}\n\\Bigg(\\prod_{i\\in\\{1,3,\\cdots 2n-1\\}}\\lim_{p_i\\cdot p_{i+1}\\to 0}p_i\\cdot p_{i+1}\\Bigg)\\mathcal{A}^{(DF)^2-photon}_{2n}\\propto&{}\n\\int \\frac{d^n\\sigma}{\\mathrm{vol[SL}(2,\\mathbb{C})]}{\\prod_i}^\\prime\\bar\\delta\\left(\\sum_{j\\neq i}\\frac{p_i\\cdot p_j}{\\sigma_{ij}}\\right)  \\mathrm{det}{}^\\prime(\\widetilde{M}_{A,2n})\\nonumber,\n\\end{align}\n\n\\noindent where the matrix $\\widetilde{M}_{A,2n}$ can be written in the following form (where $i$ and $j$ only run over the odd numbers):\n\n\\begin{align}\n\\widetilde{M}_{A,2n}^{i,j}&=\\left\\{\\begin{array}{cc}\\frac{(p_i+p_{i+1})\\cdot (p_j+p_{j+1})}{\\sigma_{ij}}&\\text{for }i\\neq j\\\\\n0&\\text{for }i=j\\end{array}\\right.,&\\widetilde{M}_{A,2n}^{i+n,j+n}&=\\left\\{\\begin{array}{cc}\\frac{p_i\\cdot p_{j+1}}{\\sigma_{ij}}&\\text{for }i\\neq j\\\\\n0&\\text{for }i=j\\end{array}\\right.,\\\\\n\\widetilde{M}_{A,2n}^{i+n,j}&=\\left\\{\\begin{array}{cc}\\frac{p_i\\cdot (p_j+p_{j+1})}{\\sigma_{ij}}&\\text{for }i\\neq j\\\\\n-\\sum_{j\\neq i}\\frac{p_i\\cdot (p_j+p_{j+1})}{\\sigma_{ij}}&\\text{for }i=j\\end{array}\\right.,&\n\\widetilde{M}_{A,2n}^{i+n,j}&=\\left\\{\\begin{array}{cc}\\frac{(p_i+p_{i+1})\\cdot p_{j+1}}{\\sigma_{ij}}&\\text{for }i\\neq j\\\\\n-\\sum_{i\\neq j}\\frac{(p_i+p_{i+1})\\cdot p_{j+1}}{\\sigma_{ij}}&\\text{for }i=j\\end{array}\\right..\\nonumber\n\\end{align}\n\nBy comparing with the formula for Einstein gravity \\eqref{Einstein Gravity}, one sees that this is the amplitude of $n$ gravitons with momenta $p_i+p_{i+1}$ where the polarization vectors have been replaced by $p_i^{(\\mu}p_{i+1}^{\\nu)}$. This makes it clear also from the quantum field theory perspective that the $(DF)^2$ photon couples to Einstein gravity.\n\n\n\n\n\n\n\n\n\n\\subsection{Connection to Witten's twistor string}\n\n\n\nEven though we only discuss bosonic states in this paper, it should be said that the spectrum of the $(\\Psi, J)$ ambitwistor string theory also contains fermions and is in fact supersymmetric  --- see  \\cite{me&renann2} for a description in the pure-spinor context. In ten dimensions, the gauge sector corresponds to SYM, while the gravity sector must be equivalent to the $R^2$ conformal supergravity studied by de Roo in \\cite{deRoo:1991at} --- see also \\cite{Bergshoeff:1982az} ---, since the action presented in that paper is supposed to be unique. \n\n\n\nFrom our point of view, it is then natural to interpret this theory as a $D$-dimensional generalization of Witten's twistor string theory \\cite{Witten:2003nn}. In four dimensions, the gauge sector describes $\\mathcal{N}=4$ SYM, while the gravity sector reduces to the conformal supergravity sector analyzed by Berkovits and Witten \\cite{Berkovits:2004jj}. Indeed, the CHY formula \\eqref{CHY CG} can be obtained from the gravity sector of this ambitwistor theory.\nNote also that a massless 3-form has no propagating degrees of freedom in four dimensions. In summary, we have the following table of approaches to the same theory:\n\n\\begin{table}[h!]\n\\centering\n\\begin{tabular}{c|c|c}\nDouble-copy &  Ambitwistor &  in $D=4$    \\\\\n\\hline \n \\hline \n   (DF)${}^2$ $\\otimes$ SYM  &  $(\\Psi,0)$                      & Berkovits--Witten sector\\\\ \\hline \n ((DF)${}^2$ $+$ $\\phi^3$) $\\otimes$ SYM  &  $(\\Psi,J)$     &  Witten's twistor string\n\\end{tabular}\n\\end{table}\n\\noindent where $\\phi^3$ stands for the bi-adjoint scalar theory, whose amplitudes can be obtained in the CHY representation through the $(J,\\tilde{J})$ ambitwistor string. It would be very interesting to obtain a more direct relation between the heterotic ambitwistor string and the twistor string studied by Berkovits and Witten, for example at the level of vertex operators. We plan to address this question in future work.\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\\section{Conclusions}\n\nIn this paper, we introduced three new, elegant CHY-type formulae and provided an ambi\\-twistor string interpretation for each of them. The string actions are all of the type $(S_L,0)$ so, together with the bosonic ambitwistor string, they form an entire new row\/column in the matrix of possible ambitwistor models.\n\nFirst we considered the $(DF)^2$ theory introduced in \\cite{conformal}. The CHY formulation of this theory is simple and exposes a property of the amplitudes that is far from obvious from the Feynman diagram perspective, namely the absence of $\\varepsilon_i \\cdot \\varepsilon_j$ terms.\n\nThe second theory we considered was an $R^2$ theory of gravity which in $D=4$ becomes conformal gravity. Our work can therefore be seen as a $D$-dimensional generalization of the paper \\cite{Berkovits:2004jj} by Berkovits and Witten, and our CHY formulation of the amplitudes does in fact reduce to their result in the appropriate limit.\n\nFinally, we looked at the $(DF)^2$-photon theory. This theory arose naturally from our studies of the previous two theories. An interesting feature of this theory is that the photon couples to regular Einstein gravity. This may seem surprising since the theory can be described using the KLT relations as the product of the non-linear sigma model and  the previously mentioned $(DF)^2$ theory. Simplistically one would expect to get at most spin-1 fields running around in such a theory since the non-linear sigma model contains only scalars and the $(DF)^2$ theory consists of scalars and gluons. This expectation is however wrong and, as demonstrated in section \\ref{s_l,0 afsnit}, one can in fact get an $n$-point Einstein gravity amplitude from the appropriate limit of an amplitude of $2n$ $(DF)^2$-photons. It will be interesting to study this theory further and try to understand this in detail. Central to this surprising fact are certainly the scalars in the $(DF)^2$ theory and their unusual color structure.\n\nThe role of the scalars is in general interesting, if somewhat mysterious. They are essential for the $(DF)^2$ theory to satisfy the color-kinematics duality, but their strange color structure leads to non-planar diagrams making contributions to tree-level amplitudes. For instance this means that in the four-point amplitudes, the numerator $n_s$ could get a term proportional $1\/u$ (terms like this can of course be removed through redefinitions of the numerators, but only in exchange for similarly weird terms in the other numerators). This in turn makes the interpretation of the function of the fields in the double copy a bit hazy, because it means that an internal graviton carrying momentum $p_1+p_2$ somehow is the product of a gluon with the same momentum and a scalar carrying momentum $p_1+p_3$. Perhaps a closer look at the amplitudes of the scalars will provide some answers. It should be fairly straightforward to get some of the amplitudes from the ${\\rm Tr}(LRLR)$-terms arising in the factorization limit as described towards the end of section \\ref{fact sub}.\n\n\n\n\n\n\n\n\n\n\\subsection*{Acknowledgments}\nWe are grateful to Henrik Johansson for suggesting the problem, sharing details about his work with Josh Nohle and for providing comments on the draft. TA acknowledges financial support from\nthe Knut and Alice Wallenberg Foundation under grant 2015.0083. OTE is supported by the Knut and Alice Wallenberg Foundation under grant KAW~2013.0235.\n\n\n\n\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
{"text":"\\section{Introduction}\nAutomatic question generation has recently \nreceived increasing attention in the natural language processing (NLP) research community.\nThe task is to generate proper questions from a given sentence or paragraph,\nwhich has many applications in NLP, including generating questions \nfor reading comprehension materials,\nand developing dialog systems for building chat robots \n\\cite{DBLP:journals\/corr\/MostafazadehMDZ16}. \nMoreover, as a reverse task of question answering, \nquestion generation can also be used to produce \nlarge scale question-answer pairs \nto assist question answering in NLP research.\n\nMany previous works have used heuristic rule based methods to tackle\nquestion generations \\cite{Rus2010},\nwhich have high requirements on the rule designs.  \nThe work in \\cite{Heilman:2011:AFQ:2520603} proposed \nto compare generated candidate questions with a ranking algorithm and induce more suitable questions. \nThe approach however depends on manually created features. \nA more recent work in \\cite{DBLP:journals\/corr\/DuSC17} proposed a neural question generation model \nto automatically induce useful representations of input sentences or paragraphs and generate\nsuitable questions with LSTM networks. \nHowever given a limited size of annotated training data, sometimes this neural model could fail\nto generate proper questions that are more suitable for the original inputs. \n\n\nIn this paper, we propose a new adaptive copying neural network (ACNN) \nmodel to tackle\nthe drawbacks of the previous works and generate proper questions.  \nThe proposed model exploits a bidirectional LSTM network with global attention mechanism \nto encode the sequential semantic information of the input sentence or paragraph.\nWhen generating semantic questions with a LSTM decoder, \nit further incorporates a copying mechanism component \nto allow more suitable and natural words to be properly generated from\nthe source input sequence in a data adaptive manner. \nWe conduct experiments on the most widely used Stanford Question Answering Dataset (SQuAD) \\cite{DBLP:journals\/corr\/RajpurkarZLL16}. The empirical results show the proposed ACNN model can outperform \nthe state-of-the-arts in terms of BLEU-$n$ and ROUGE-$L$ scores. \n\n\\begin{figure*}[t]\n\\centering\n\\includegraphics[width=0.9\\textwidth]{ACNN}\\\\\n\\caption{The proposed Adaptive Copying Neural Network (ACNN). }\n\\label{fig1}\n\\end{figure*}\n\\section{Related Work}\n{Question generation} (QG) \nhas drawn a lot attention in the recent years. \nThe previous work in \\cite{article} applied minimal recursion semantics (MRS) to represent the meaning of sentences and then transfer MSR to questions. \nThe work in \\cite{Heilman:2011:AFQ:2520603} proposed an overgenerate-and-rank approach \nto generate and select high quality questions via ranking.\nThe authors of \\cite{Chali:2015:TTG:2812180.2812181} focused on generating questions based on topics. \nThe work in \\cite{deep-questions-without-deep-understanding} used ontology crowd-sourcing to encode \nthe original text in an ontology and align the question templates to select the most relevant ones.\nThe work in \\cite{DBLP:journals\/corr\/SerbanGGACCB16} \ntook the knowledge based information as input \nand generated questions based on it,\nwhile  \n\\cite{DBLP:journals\/corr\/ZhouYWTBZ17} combined the answer position of the text.\nThe recent work in \\cite{DBLP:journals\/corr\/DuSC17} proposed a neural question generation model \nbased on LSTMs which demonstrates good empirical results.\n\n\n{Question answering} (QA) \nis a reverse task of QG. It shares similarities and sometimes mixed together with QG.\nThe works in \\cite{DBLP:journals\/corr\/XiongZS16} and \\cite{DBLP:journals\/corr\/ChenBM16a} \nused similar neural networks with attention mechanisms. \nThe work of \\cite{Song:2017:SAN:3018661.3018704} focused on retrieving non-factoid community questions as well as the lengths of the answers, while the work \n\\cite{DBLP:journals\/corr\/YangHSC17} took reading comprehension as main tasks and question answering as auxiliary tasks. \nThe authors of \\cite{NIPS2016_6469} address QG and QA simultaneously to boost both of them.\n\n\\section{Model}\nGiven an input sentence or paragraph ${\\bf x}$ which is a sequence of tokens \n$[x_1,\\cdots, x_N]$, we aim to generate a natural question ${\\bf y}=[y_1,\\cdots,y_{|{\\bf y}|}]$ from it.\nInspired by the work of\n\\cite{DBLP:journals\/corr\/DuSC17}, \nwe use a bidirectional long short-term memory network (LSTM)\nwith global attention mechanism\nto perform automatic question generation. \nWe add a copying mechanism onto this neural model \nto incorporate\noriginal input vocabulary information in the decoding phase to generate proper questions. \nThe proposed ACNN model is demonstrated in Figure~\\ref{fig1}.\nThis end-to-end learning model has two fundamental parts, attention based encoder and \ncopying mechanism based decoder. \n\n\n\\subsection{Attention Based Encoder}\n\nWe use a bidirectional LSTM \nto encode the given sequence of tokens in the input sentence \n${\\bf x}$. \nLet $\\overrightarrow {{\\bf h}_t}$ denote the hidden state at time step $t$ for the forward LSTM \nand  $\\overleftarrow {{\\bf h}_t}$ for the backward LSTM. \nThe bidirectional LSTM produces the hidden states as follows:\n\\[\n\\overrightarrow {{\\bf h}_t} = \\overrightarrow{LSTM} (x_t, \\overrightarrow{{\\bf h}_{t-1}})\n\\]\n\\[\n\\overleftarrow {{\\bf h}_t} = \\overleftarrow{LSTM} (x_t, \\overleftarrow{{\\bf h}_{t+1}})\n\\]\nBy concatenating the hidden states\nfrom both directions \nwe have the following context dependent hidden representation at step $t$ \n${\\bf h}_t=[\\overrightarrow {{\\bf h}_t};\\overleftarrow {{\\bf h}_t}]$. \n\nThe attention based encoding of ${\\bf x}$ at a decoding time step $k$ is then computed as\na weighted average of the representation vectors across ${\\bf h}_t$,\n\\[\n{\\bf c}_k = \\sum_{t=1}^N a_{k,t} {\\bf h}_t.\n\\]\nThe attention weights $\\{a_{k,t}\\}$ are calculated using a softmax normalization \n\\[\na_{k,t} = \\frac{\\exp(e_{k,t})}{\\sum_{j=1}^N \\exp(e_{k,j})},\n\\]\n\\[\ne_{k,t} = \\tanh({\\bf d}_k^\\top W_h {\\bf h}_t) \n\\]\nwhere $W_h$ is the model parameter to be learned, and\n${\\bf d}_k$ is the hidden decoding state at time step $k$ \nwhich we will introduce below.\n\nWe also consider encoding\nthe truncated paragraph (with length $L$) that contains sentence ${\\bf x}$ using the bidirectional \nencoding LSTM \nto replace the encoding of ${\\bf x}$.\n\n\n\\subsection{Copying Mechanism Based Decoder}\nThe decoding process is to generate question ${\\bf y}$ from the given sentence ${\\bf x}$,\nwhich is a probabilistic sequence prediction that can be factorized as:\n\\begin{eqnarray}\nP({\\bf y}|{\\bf x}) = \\prod_{k=1}^{|{\\bf y}|}P(y_k | y<k, {\\bf x})\n\\end{eqnarray}\nThis is also the empirical probability we need to maximize in the training process across \nall the annotated training instances.\n\nWe compute the local conditional word output probability \n$P(y_k | y<k, {\\bf x})$ by integrating both a LSTM attention-based decoding component\nand a copying mechanism component \nsuch that \n\\begin{eqnarray}\n&P(y_k | y<k, {\\bf x}) \n\\nonumber\\\\\n= &z_k P_{cop}(y_k) + (1-z_t)P_{att}(y_k) \n\\end{eqnarray}\nThe attention part\n$P_{att}(y_k )$ generates words from\nthe common decoder vocabulary, \nand it is computed \non the attention vector ${\\bf c}_k$ and the hidden\nstate vector ${\\bf d}_k$ from a decoding LSTM:\n\\[P_{att}(y_k)=softmax(W_y\\tanh(W_k[{\\bf d}_k;{\\bf c}_k]))\\] \nwhere $W_y$ and $W_k$ are model parameters.\n\nThe copying mechanism component $P_{cop}(y_k)$ \ngenerates (copies) words from the individual vocabulary of the source input sequence.\nWe compute it as\n\\begin{eqnarray}\nP_{cop} (y_k) = softmax(\\!V^\\top\\! (V [{\\bf d}_k;\\! {\\bf c}_k]\\!+\\!b_1)\\!+\\!b_2)\n\\end{eqnarray}\nwhen $y_k$ is from the unique word set of the source input sequence,\nwhere $V, b_1$ and $b_2$ are model parameters.  \nSuch a copying mechanism can help \nincorporating words from the original data into the generated questions.\n\n\nThe combination weight $z_t$ is \nthe switch for deciding generating the word from the vocabulary or \ncopying it from the input sequence. \nWe\ncalculate $z_t$ as follows:\n\\begin{eqnarray}\nz_k = \\sigma (W_d^\\top {\\bf d}_k + W_c^\\top  {\\bf c}_k + W_s^\\top y_{k-1} + b)\n\\end{eqnarray}\nwhere $W_d$, $W_c$, $W_s$ and $b$ are model parameters,\nand $\\sigma$ denotes a sigmoid function.\nHence $z_k$ functions as a selection gate that makes data adaptive selection\nbetween the attention component and the copying component. \n\\section{Experiments}\n\nWe conducted experiments on the widely used Stanford Question Answering Dataset (SQuAD) \\cite{DBLP:journals\/corr\/RajpurkarZLL16}. \nWe used the version released by \\cite{DBLP:journals\/corr\/DuSC17}.\nIt was split into three parts -- training set, developing set and test set.\nThe training set contains 70,484 input-question pairs, \nthe development set contains 10,570 input-question pairs, \nand the test set contains 11,877 input-question pairs. \n\n\\begin{table*}[t]\n\\centering\n\\caption{The comparison results in terms of BLEU and ROUGE scores. \nThe best scores in baselines and ACNN are highlighted using boldface.\\\\}\n\\label{tab1}\n\\begin{tabular}{l|cccc|c}\n\\hline\nModel&BLEU-1&BLEU-2&BLEU-3&BLEU-4&ROUGE-$L$\\\\\n\\hline\nSeq2Seq&31.34&13.79&7.36&4.26&29.75\\\\\nDu-sent&\\textbf{43.09}&\\textbf{25.96}&\\textbf{17.5}&\\textbf{12.28}&\\textbf{39.75}\\\\\nDu-para&42.54&25.33&16.98&11.86&39.37\\\\\n\\hline\nACNN-sent &\\textbf{44.78}&\\textbf{26.83}&\\textbf{18.72}&\\textbf{13.97}&\\textbf{41.08}\\\\\nACNN-para&44.37&26.15&18.02&13.49&40.57\\\\\n\\hline\n\\end{tabular}\n\\end{table*} \n\\begin{table*}[t]\n\\centering\n\\caption{The results in terms of BLEU and ROUGE scores with different paragraph lengths. The best scores are highlighted using boldface.\\\\}\n\\label{tab2}\n\\begin{tabular}{l|cccc|c}\n\\hline\nModel&BLEU-1&BLEU-2&BLEU-3&BLEU-4&ROUGE-$L$\\\\\n\\hline\nACNN-para-150&43.97&25.63&17.48&12.91&39.95\\\\\nACNN-para-120&44.22&25.94&17.80&13.26&40.33\\\\\nACNN-para-100&\\textbf{44.37}&\\textbf{26.15}&\\textbf{18.02}&\\textbf{13.49}&\\textbf{40.57}\\\\\n\\hline\n\\end{tabular}\n\\end{table*}\n\n\\paragraph{Experimental setting}\nThe proposed model is built using Torch 7 on the OpenNMT system\n\\cite{DBLP:journals\/corr\/KleinKDSR17}. \nWe adopted the same setting as \n\\cite{DBLP:journals\/corr\/DuSC17}.\nWe kept the most frequent 45K tokens as the encoder vocabulary and \n28K tokens as the decoder vocabulary. \nWe used the word embeddings\nreleased by \\cite{pennington2014glove} as pre-training embeddings of the input words. \nWe set the size of all LSTM hidden state vectors as 600 and the number of LSTM layers as 2. \nFor the paragraph encoder, we set the length of paragraphs as 100. \nWe use dropout technique in \\cite{Srivastava:2014:DSW:2627435.2670313} with probability $p$ = 0.3. \nDuring testing, we set the beam search size to 3.\nWe used stochastic gradient descent(SGD) as optimization algorithm with initial learning rate $\\alpha$ = 1.0 \nand halve it when at epoch 8. To speed up training, we set mini-batch size to 64.\n\n\\paragraph{Comparison methods}\nWe compared the proposed ACNN model with the following methods on question generation.\n(1) {\\em Seq2Seq}: This model was proposed by \\cite{NIPS2014_5346} \nwhich is a basic sequence to sequence transformation model. \n(2) {\\em Du-sent}: This is the state-of-the art model developed in \\cite{DBLP:journals\/corr\/DuSC17},\nand we use it with the sentence level encoder and pre-trained word embeddings. \n(3) {\\em Du-para}: This denotes the same model as above but with paragraph level encoder to be incorporated. \nFor our proposed ACNN model, we also tested two versions, \n{\\em ACNN-sent} with only sentence encoder and\n{\\em ACNN-para} with only paragraph encoder. \n\n\n\\paragraph{Evaluation metrics}\nWe used two types of commonly used evaluation metrics, BLEU-$n$ and ROUGE-$L$, to evaluate the testing results.\nBLEU-$n$ (Bilingual Evaluation Understudy) \\cite{Papineni:2002:BMA:1073083.1073135}\nis a score that uses $n$ grams to calculate the correspondence \nbetween the machine generated output and the ground truth. \nROUGE (Recall-Oriented Understudy for Gisting Evaluation) \n\\cite{rouge-a-package-for-automatic-evaluation-of-summaries} \nmeasures the co-occurrences \nbetween the system-generated summary and the content in a human-generated summary. \nROUGE-$L$ measures the co-occurrences of the longest common subsequence. \nThe higher scores in these two metrics indicate better performance. \n\n\n\\subsection{Experimental Results}\n\nThe experimental results \nin terms of BLEU$1-4$ and ROUGE-$L$ scores \nfor all the comparison methods\nare reported in Table \\ref{tab1}. \nThe best results among the comparison baselines and the proposed ACNN variants \nare highlighted using boldface font separately.\n\n\nWe can see that among the comparison methods, \n\\textit{Du-sent} produced the best performance \nin terms of all the evaluation metrics. \n\\textit{Du-para}, though incorporated paragraph, produced slightly inferior performance.  \nBoth {\\em Du-sent} and {\\em Du-para} greatly outperform {\\em Seq2Seq}.\nAmong the two variants of our proposed model, {\\em ACNN-sent} slightly outperforms \n{\\em ACNN-para} which uses paragraph as inputs.\nThis is consistent with the {\\em Du-}methods and might be caused by the noise in the paragraphs.\nNevertheless, both variants of the proposed ACNN model consistently outperform the other comparison methods in terms of \nall the evaluation metrics. \nThe comparison results between {\\em ACNN-sent} and {\\em Du-sent} validate the effectiveness of \nincorporating the copying mechanism into the bidirectional LSTM question generation model.\n\n\n\\subsection{Impact of Paragraph Length}\nWe also investigated the impact of the paragraph length on the performance \nof the proposed variant {\\em ACNN-para}.  \nWe tested three different length values, 100 (the default value used above), 120 and 150. \nThe comparison results are reported in \nTable \\ref{tab2}.\nWe can see that when increasing the paragraph length, the test scores decrease.\nThis validates our analysis above on paragraph introducing noise:\nAlthough longer paragraphs contain more contextual information, they include more irrelevant noisy information as well. \n\\section{Conclusion and Future Work}\nIn this paper, we proposed an adaptive copying neural network (ACNN) model for question generation. \nWe incorporated a copying mechanism component into a bidirectional LSTM model with global attention mechanism to improve its capacity on \ngenerating proper natural questions. \nWe conducted experiments on the widely used \n{\\em SQuAD} dataset and showed the proposed model outperforms the state-of-the-art method in the literature \nin terms of two types of evaluation metrics, \nBLEU-$n$ and ROUGE-$L$. \n\nIn the future, we plan to extend the proposed model \nto address multi-task question generation on multi-documents with similar topics,\naiming to generate questions with similar copying mechanism that are consistent with human brain activities.  \nThe copying mechanism can also be expected to allow model adaptation across domains.\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
{"text":"\n\n\n\\section{Introduction}\n\nThe capability of harvesting energy has been frequently stated as a measurement of the technological advancement of civilizations, most famously by Nikolai Kardashev~\\cite{kardashev1964transmission}, after which the Kardashev-scale has been named. A type-II civilization for example is defined as a civilization that is capable of utilizing the complete energy output of its home star, which would be roughly $4 \\cdot 10^{26}$ J\/s in case of our own sun. A swarm of solar harvesting satellites orbiting the star in dense formation has been proposed as enabling technology for such an ambition. \n\nWhile this idea has been explored in science fiction literature before, its first scientific treatment is attributed and named after Freeman Dyson~\\cite{dyson1960search}. Dyson's goal was to show the physical plausibility of a giant energy harvesting biosphere by calculations based on our own solar system, leading to important implications with regards to the search for extraterrestrial life. While engineering aspects and astrodynamic specifics were not discussed in Dyson's original work, his thought experiment sparked the imagination of countless minds and served as inspiration for the Global Trajectory Optimization Competition (GTOC) in its 11th edition~\\cite{Shen2022}.\n\nThe setup of this challenge tasked its participants with the assembly of a precursor structure, a Dyson ring, consisting of twelve equally distributed stations on a circular orbit situated in our solar system. An ensemble of ten mother ships needs to be launched from Earth towards the outer solar system to collect the necessary mass for construction. This collection is performed through the deployment of a multifunctional asteroid transfer device (ATD), which serves as an abstraction for the future technical systems required to effectively mine and transport the resources of such a body. Once an ATD gets activated, part of the asteroid's own mass will be continuously converted into constant acceleration, allowing it to travel towards the construction orbit. Synchronizing the arrival of such asteroids well-phased within a narrow time interval for building while maintaining an optimal arrival mass presents a complex and unique scheduling challenge, including trade-offs between the construction orbit, activation times, mass distribution among stations and combinatorial selection of suitable candidate asteroids. While the actual construction of megastructures at the envisioned scale of the GTOC~11 seems still out of reach, the basic concepts, for example, of asteroid mining~\\cite{andrews2015defining, hellgren2016asteroid} and solar powered satellites~\\cite{summerer2005advanced, flournoy2011solar} are regarded as emerging space technologies, that will enable us to exploit extraterrestrial resources on an unprecedented scale.\nMoreover, the challenging planning and scheduling aspects of large-scale space constructions provide a compelling environment for which multiple different optimization techniques are required.\n\nOn a high level, any solution strategy to the GTOC~11 challenge needs to address the sub-challenges of 1. asteroid selection, 2. station orbit determination and 3. construction time scheduling. The asteroid selection sub-challenge can be seen as a combinatorial problem that requires the selection of favorable sequences of asteroids linked by optimal transfers. The dominant solution strategies for this type of problem are tree searches, ant colony optimization or hybridisations of both~\\cite{simoes2017multi}. An additional difficulty is introduced by the concurrency of the selection, as (ideally) coordination between all ten motherships is exploited. The station orbit determination sub-challenge is strongly tied to the solutions of the other sub-challenges as it links multiple objectives together (as we will show in our preliminary analysis of the GTOC~11 objective function). Tackling it after the asteroid selection decreases its difficulty considerably, as an optimal orbit for a small subset of asteroids can be modelled and solved as a non-linear optimization problem. While this has the downside that the asteroid selection is blind (i.e. it is not informed about the optimal station orbit) an earlier solution of this sub-challenge implies solving the GTOC~11 holistically by directly optimizing for the objective function, which is by design of the competition practically impossible. Lastly, the construction time scheduling sub-challenge is unique and difficult to compare to any established scheduling or sequencing problems that the authors are aware of. In particular, the fact that asteroids and stations move along orbits together with the tight construction constraints demand the development of entirely new algorithms, as it is in the spirit of GTOC. The purpose of this work is to describe the end-to-end optimization pipeline our team used to submit solutions during the competition: a pipeline that integrates scheduling algorithms, evolutionary optimization, machine learning and space flight mechanics routines and solves the three sub-challenges in the aforementioned order. Figure~\\ref{fig:flowchart} shows a schematic of this pipeline with references to the corresponding sections and algorithms.\n\n\\begin{figure}[ht]\n\t\\centering\n     \\includegraphics[width=.9\\textwidth]{figures\/gtocflowchart.png}\n    \\caption{Schematic of solution pipeline. When appropriate, sections and algorithms describing the procedures (yellow diamond) or intermediate results (rectangles) are given.}    \n    \\label{fig:flowchart}     \n\\end{figure}\n\nWe begin this report with a preliminary analysis of the GTOC~11 objective function, the related continuous-thrust optimal control problem and the population of candidate asteroids provided as part of the challenge setup. Next, we describe how a Lazy Race Tree Search can be designed and improved by a time estimation provided by supervised machine learning, leading to a quick generation of a large set of possible mother ship trajectories and related ensembles. The next section describes our ring building pipeline, including the determination of its orbital parameters, the optimization of the activated asteroid trajectories and the solution of related assignment problems by the Hungarian algorithm in combination with evolutionary optimization techniques. Following these conceptual explanations, our result section describes the generation of our specific submission, which enabled our team, ACT\\&Friends, to obtain the second overall rank in the final GTOC~11 leaderboard. We conclude with an outlook on the lessons learned from the competition and point out directions demanding further investigation.\n\n\\section{Preliminary Analysis}\n\n\\subsection{Objective Function}\nThe objective function, as released as performance index in the original problem description, has the form:\n\\begin{equation}\nJ = B \\cdot \\frac{10^{-10} \\cdot M_{min}}{a^2_{D} \\sum\\limits_{k=1}^{10}\\left(1 + \\Delta V_{k}^{Total} \/ 50 \\right)^2}\n\\label{eq:objfun}    \n\\end{equation}\nwhere $B$ is a time penalty factor: a number decreasing as the competition advanced and designed as to stimulate early submissions and an active leaderboard.\n$M_{min}$ is the minimum cumulative mass delivered at one of the twelve stations. This term encourages to evenly deliver asteroid mass to all target stations and to activate more asteroids.\nThe target Dyson ring radius $a_D$ appears explicitly only in the denominator, benefitting designs with small ring radii. The Dyson ring radius also factors into the determination of the asteroid mass delivered (numerator) and thus an optimal configuration exists at some unknown value.\nFinally, $\\Delta V_{k}^{Total}$ is the total velocity increment needed by each mother ship to deliver the ATDs and thus enabling asteroid activation. This term encourages to find fuel efficient trajectories in general, as well as to decrease the number of activated asteroids, assuming that significant velocity changes are required in between asteroid visits. The trade-offs and exact balances of all these contributions can only be made approaching the problem holistically, accounting for the actual asteroid orbits and spacecraft dynamics. Nevertheless, a preliminary qualitative analysis is possible by introducing a few simplifying assumptions.\n\nLet us assume that the final configuration is made by twelve stations, each receiving $N$ identical asteroids with final mass $m_{arr}(a_D)$. The mass delivered depends on the Dyson ring radius, as smaller rings will require more time to be reached resulting in a lower mass at arrival.\nWe also assume that, on average, the cost for a mother ship to make one transfer and thus activate one asteroid is $\\Delta \\tilde V$ and is uniform across the trajectories of the ten mother ships. Since the total number of asteroids to be activated is, under these definitions, $12 N$ and they are activated by ten spacecraft, it follows that each mother ship will use a total velocity increment $\\Delta V_{k}^{Total} = \\frac{12}{10} N \\Delta \\tilde V$. We can therefore rewrite the objective function, neglecting the various constants, as to highlight all of its trade-offs as:\n\\begin{equation}\nJ \\propto \\frac{N m_{arr}(a_D)}{a^2_{D} \\left(1 + \\frac{12}{10} N \\Delta \\tilde V \/ 50 \\right)^2}\n\\label{eq:objfun_simple}    \n\\end{equation}\nUsing the above equation and setting its derivative with respect to $N$ to zero, we find that:\n$$\n\\frac{\\partial J}{\\partial N} =0 \\rightarrow N^* = \\frac{41.6}{\\Delta \\tilde V}\n$$\nwhich tells us that spending, for example, $1$ km\/s on average to activate one asteroid, should result in each mother ship activating around 42 asteroids. Note that no team during the competition submitted solutions quantitatively in this regime, suggesting the impossibility to fulfill the hypothesis made here on the Dyson ring's final configuration, but maybe also indicating the possibility to further improve the trajectories found so far.\nThe expression above is interesting, nevertheless, as it states that the optimal number of asteroids delivered to each station does not depend on the chosen Dyson ring radius, but only on the fuel efficiency of the mother ship trajectories. It also reveals the various trade-offs clearly, showing how more activated asteroids (i.e. higher $N$) do not necessarily result in a better score if the cost paid by the mother ship to activate them is too high. \nAs for the Dyson ring radius, the expression also highlights how its optimal value depends exclusively on the trade-off dictated by the quantity $\\frac {m_{arr}(a_D)}{a^2_{D}}$. By introducing additional assumptions, a more detailed analysis could also take the derivative with respect to $a_D$ into account, approximating the functional form for $m_{arr}(a_D)$, for example, making use of the Edelbaum approximation \\cite{edelbaum1961propulsion}. However, in this report we will not follow this idea and move on by introducing the optimal control problem associated with the asteroid transfers.\n\n\\subsection{Low Thrust Continuous Transfers}\n\\label{subsec:low_thrust_transfers}\nThe trajectory of an asteroid after its activation is determined by a constant acceleration $\\Gamma = 10^{-4} m\/s^2$ acting along the direction $\\hat{\\mathbf i}(t)$.\nThe associated time-optimal control problem (to find $\\hat{\\mathbf i}(t)$) can be efficiently solved by either direct or indirect methods~\\cite{von1992direct}.\nAfter experimenting with both approaches, we decided to use indirect methods exclusively as they offered several advantages, e.g. their numerical precision was most of the time compliant with the error tolerances of $10$ km in position and $0.01$ m\/s demanded by the GTOC~11 problem statement.\nApplying the approach from Pontryagin's theory~\\cite{pontryagin1962maximum}, we started from the simple equations of motion in Cartesian coordinates: \n\\begin{equation}\n\\label{eq:dyn}\n\\left\\{ \n\\begin{array}{l}\n    \\dot{\\mathbf{r}} = \\mathbf{v} \\\\\n    \\dot{\\mathbf{v}} = -\\frac{\\mu}{r^3}\\mathbf{r} +\\Gamma\\mathbf{\\hat{i}}(t)\n\\end{array}\n\\right.\n\\end{equation}\nwhere $\\mu$ is the gravitational constant of the Sun. We thus considered the problem of finding $t_f$, $t_f-t_0$ and a function $\\mathbf{\\hat{i}}(t)$ with $t \\in [t_0, t_f]$ so that, under the dynamics defined in Eq.(\\ref{eq:dyn}), the state is steered from the initial state $\\mathbf r_{ast}(t_0), \\mathbf v_{ast}(t_0)$ representing some asteroid ephemerides in $t_0$ to the target state $\\mathbf r_{st}(t_f),\\mathbf v_{st}(t_f)$ representing the target station position and velocity. \nThe following cost function needs to be minimal:\n$J = t_f-t_0 = \\int^{t_f}_{t_0} 1 dt$, leading to the introduction of the Hamiltonian: \n\n\\begin{equation}\n\\label{eq:hamiltonian}\n    \\mathcal H(\\mathbf{r},\\mathbf{v},\\boldsymbol{\\lambda}_{\\mathbf r},\\boldsymbol{\\lambda}_{\\mathbf v},\\mathbf{\\hat{i}}) = \\boldsymbol{\\lambda}_{\\mathbf r}\\cdot \\mathbf{v}+\\boldsymbol{\\lambda}_{\\mathbf v}\\cdot\\bigg(-\\frac{\\mu}{r^3}\\mathbf{r}+\\Gamma\\mathbf{\\hat{i}}\\bigg)+1\n\\end{equation}\nwhere we have introduced the co-state's functions $\\boldsymbol{\\lambda}_{\\mathbf r}$ and $\\boldsymbol{\\lambda}_{\\mathbf v}$. It is therefore straightforward to obtain the following consequence of the Pontryagin's maximum principle:\n\n\\begin{equation}\n\\label{eq:duh}\n\\mathbf{\\hat{i}}=-\\frac{ \\boldsymbol{\\lambda}_{\\mathbf v}}{\\lambda_v}\n\\end{equation}\nwhich defines the optimal direction of the acceleration, and the corresponding augmented dynamics derived from the Hamiltonian performing the derivatives $\\dot {\\mathbf x} = \\frac{\\partial \\mathcal H}{\\partial \\boldsymbol \\lambda}$, $\\dot {\\boldsymbol \\lambda} = - \\frac{\\partial \\mathcal H}{\\partial \\mathbf x}$:\n\\begin{equation}\n\\label{eq:augdyn}\n\\left\\{ \n\\begin{array}{l}\n    \\dot{\\mathbf{r}} = \\mathbf{v} \\\\\n    \\dot{\\mathbf{v}} = -\\frac{\\mu}{r^3}\\mathbf{r}-\\Gamma\\frac{ \\boldsymbol{\\lambda}_{\\mathbf v}}{\\lambda_v}\\\\\n    \\dot{\\boldsymbol\\lambda}_{\\mathbf{r}} = \\mu \\left(\\frac{\\boldsymbol{\\lambda}_{\\boldsymbol v}}{r^3} - 3(\\boldsymbol \\lambda_{\\boldsymbol v}\\cdot\\mathbf r)\\frac{\\mathbf r}{r^5} \\right)  \\\\\n    \\dot{\\boldsymbol\\lambda}_{\\mathbf{v}} = - \\boldsymbol{\\lambda}_{\\mathbf r}\n\\end{array}\n\\right.\n\\end{equation}\nAccording to Pontryagin's theory, an optimal transfer will necessarily be a solution to the above differential equations with the added condition $\\mathcal H=0$ as we are also considering a free time problem. \nNote however that this last condition is not necessary since we can always multiply the co-states for some coefficient $\\lambda_0$ and get a new equivalent solution resulting in $\\mathcal H=0$ since all the relevant equations are homogeneous in the co-states. \nWe will nevertheless seek also a zero Hamiltonian solution as to avoid numerical instabilities. \nConsider now the shooting function:\n\n\\begin{equation}\n    \\Phi_{t_f}(t_0, \\boldsymbol{\\lambda}_{\\mathbf{r0}},  \\boldsymbol{\\lambda}_{\\mathbf{v0}}) = [\\mathbf r(t_0+T) - \\mathbf r_{st}(t_f), \\mathbf v(t_0+T) - \\mathbf v_{st}(t_f), \\mathcal H]\n\\end{equation}\n\nFor any given $t_f$, assuming $t_f = t_0+T$, the roots of the above function allow us to find the optimal transfer for an asteroid. Clearly, if the resulting $t_0$ is before the activation date, there is no valid solution arriving in $t_f$ at the chosen station. Formally, when looking for optimal asteroid transfers to a given station, we thus solve the following rendezvous optimization problem:\n\n\\begin{equation}\n\\label{eq:rdvz}\n    \\begin{array}{rl}\n        \\mbox{find:} &  t_f, t_0, \\boldsymbol{\\lambda}_{\\mathbf{r0}},  \\boldsymbol{\\lambda}_{\\mathbf{v0}}\\\\\n        \\mbox{to minimize:} & t_f-t_0\\\\\n        \\mbox{subject to:}&  t_f = T + t_0 \\\\\n        &  \\Phi_{t_f}(t_0, \\boldsymbol{\\lambda}_{\\mathbf{r0}},  \\boldsymbol{\\lambda}_{\\mathbf{v0}}) = \\mathbf 0 \\\\\n        & t_0 > t^{act} \\\\\n        & t_f \\in [\\underline t_f, \\overline t_f]\n    \\end{array}\n\\end{equation}\n\nwhere an arrival window $[\\underline t_f, \\overline t_f]$ is assumed, and $t^{act}$ indicates the asteroid activation epoch.\nSimilarly, when considering a phase-free problem (for example for the Edelbaum approximation learning), we considered the following version where the time of flight $T$ is no longer linked to the starting and arrival position (a simple numerical trick to get the phase-free version of the same problem):\n\n\\begin{equation}\n\\label{eq:phasefree}\n    \\begin{array}{rl}\n        \\mbox{find:} &  T, t_f, t_0, \\boldsymbol{\\lambda}_{\\mathbf{r0}},  \\boldsymbol{\\lambda}_{\\mathbf{v0}}\\\\\n        \\mbox{to minimize:} & T\\\\\n        \\mbox{subject to:}\n        &  \\Phi_{t_f}(t_0, \\boldsymbol{\\lambda}_{\\mathbf{r0}},  \\boldsymbol{\\lambda}_{\\mathbf{v0}}) = \\mathbf 0 \n    \\end{array}\n\\end{equation}\n\nIt is worth mentioning that we experimented also in writing the shooting function in terms of equinoctial elements instead of Cartesian, observing an improvement in the convergence of the sequential quadratic programming method we employed to solve the problems. \nAlthough comparing computational times for numerically solving specific classes of problems is arguably involved due to the dependencies related to the deployed hardware and software on top of the problem specific conditions and parameters, a rough indication on the order of magnitude can be made. One instance of either problem defined above is solvable on a modern single threaded CPU in one second on average. Note that the guess provided on the initial value of each of the co-states is here a uniform random number in the bounds [-10,10].\nSome local minima are present, though, and therefore multiple starts may provide better solutions.\nThis relatively short computational time allowed us to consider strategies where solving large numbers of such problem instances is obligatory.\nOn a more technical note, we used a Taylor-based method (Heyoka~\\cite{biscani2021revisiting}) for all numerical propagation of the augmented dynamical equations and SNOPT~\\cite{gill2005snopt} as implementation of the SQP method for solving the optimization problems transcribed as a non-linear program (NLP) using the pagmo\/pygmo package~\\cite{biscani2020parallel}.\n\n\n\\subsection{Asteroids}\nThe core data for the competition problem lies in the provided dataset of $83\\,453$ asteroids, defining the principal sources of mass available for transport to the twelve ring stations. In addition to the orbital elements specifying the asteroids' orbits, each asteroid has an initial mass $m_0$ between $2.266 \\cdot 10^{9}$ and $m_{max} = 1.963 \\cdot 10^{14}$ kg. During its constant acceleration transfer to the target ring, the mass of an asteroid decreases proportionally to its initial mass by a factor of $\\alpha = 6 \\cdot 10^{-9}$. Consequently, the longer the asteroid needs to reach its final destination, the less of its initial mass will remain for ring construction. Given a transfer time of $T$ and target semi-mayor axis $a_D$, we define the arrival mass $m_{arr}$ of an asteroid as\n\\begin{equation}\n\\label{eq:arr_mass}\nm_{arr}(a_D) = m_0 \\cdot (1 - \\alpha \\cdot T).\n\\end{equation}\n\n\\begin{figure}[ht]\n\t\\centering\n     \\includegraphics[width=.9\\textwidth]{figures\/asteroid_population.png}\n    \\caption{Detailed relationships between critical orbital elements, (initial) mass and arrival mass at the optimal station assuming a Dyson radius of $1.29$ AU at $0^\\circ $ inclination. Only asteroids arriving with at least $10^9$ kg are shown ($N = 79,266$).}    \n    \\label{fig:asteroid_population}     \n\\end{figure}\n\nThus, initial mass and orbital elements are crucial for the asteroid selection. Solving the optimal control problem (OCP) as given by Eq.(\\ref{eq:phasefree}) for all asteroids to a hypothetical Dyson ring at $1.29$ AU and $0^\\circ$ inclination gives a first indication of the mass we can expect to arrive at this particular Dyson ring orbit, shown in Figure~\\ref{fig:asteroid_population} for all asteroids that arrive with at least $10^9$ kg ($N = 79,266$). Notably, the semi-major axis of these asteroids features a trimodal distribution with peaks at $\\approx 2.33$, $2.67$ and $3.15$ AU. Asteroids at larger distances lead to a reduced arrival mass given their prolonged transfer times.\nEccentricity and inclination are on average limited for most asteroids ( $0.150 \\pm 0.0783 $ and $ 9.21^\\circ \\pm 6.14^\\circ$, respectively) with higher eccentricities and inclinations also leading to a lower arrival mass. The latter, however, is a slightly biased result given our assumption of the $0^\\circ$ inclination of the target Dyson ring.\nThe initial mass distribution seems to be independent of semi-major axis, inclination and eccentricity, with an average of $6.71 \\pm 5.21 \\cdot 10^{13}$ kg. The majority of asteroids feature a smaller mass with only $25.9$\\% having a mass greater than $10^{14}$ kg. The arrival mass is on average $1.52 \\pm 1.38 \\cdot 10^{13}$ kg and shows a clear relationship to all other displayed parameters and its distribution has a higher kurtosis compared to the initial mass distribution. Only $2.8$\\% of all asteroids have an arrival mass larger than $5 \\cdot 10^{13}$ kg.\n\nIn summary, we deduce the following insights from our analysis: First, having a method to reliably estimate transfer time $T$ and thus arrival mass $m_{arr}(a_D)$ is essential for any high level strategy of asteroid selection. Second, the dataset of asteroids can be reduced considerably if focus is given to the long tail of asteroids providing higher arrival masses. Finally, constructing a Dyson ring of small radius (e.g. close to the minimum possible of $0.65$ AU) results in a relatively poor payoff, as already at $1.29$ AU further out asteroids bring in comparatively little mass.\n\n\\section{Mother Ship Trajectory Design}\n\n\\subsection{Earth-to-Asteroid and Asteroid-to-Asteroid Transfers}\n\\label{subsec:a2a}\n\nThe task of each mother ship is to depart from Earth and deploy a number of ATDs (Asteroid Transfer Devices) within a limited mission time of $20$ years. As such, the Earth-to-Asteroid (E2A) and the Asteroid-to-Asteroid (A2A) transfers are basic building blocks for the assembly of a trajectory. While the problem statement allows for up to four impulsive velocity changes between transfers, our search strategy is based on the rapid optimization of three-impulse legs, containing a departure, deep space and arrival impulse. The three impulse strategy was deemed, at this point, to be a good compromise between computational complexity and optimality of the resulting transfers. In hindsight, since most optimal transfers requested anyway only two impulses, the whole mother ship design could have been simplified considerably.\n\nFor any given target asteroid, the E2A optimization problem is defined as follows:\n\n\\begin{equation}\n    \\begin{array}{rl}\n        \\mbox{find:} &  t_0, T, V_{\\inf}, u, v, \\eta\\\\\n        \\mbox{to minimize:} & \\Delta V\\\\\n    \\end{array}\n\\end{equation}\nwhere $t_0$ is the starting epoch, $T$ the transfer time, $ V_{\\inf}$ is the relative spacecraft velocity at departure and $u$ and $v$ are angles indicating the direction of departure. Following the competition setup, $V_{\\inf}$ is discounted by $6$km\/s but can be higher to allow for faster transfer times. After this departure impulse, the mother ship is propagated for $\\eta T$ towards its initial direction until another impulse can be applied (deep space maneuver). From there, a Lambert arc connecting to the target asteroid is constructed and the final impulse is adjusted to match the asteroid's velocity up to $2$km\/s for the deployment of the ATD.\n\nFollowing the deployment of the first ATD, the starting epoch, position and initial velocity of the spacecraft is fixed for the following transfer. Thus, the A2A optimization problem has one less decision variable, but is otherwise identical:\n\n\\begin{equation}\n\\label{eq:a2a}\n    \\begin{array}{rl}\n        \\mbox{find:} &   T, V_{\\inf}, u, v, \\eta\\\\\n        \\mbox{to minimize:} & \\Delta V\\\\\n    \\end{array}\n\\end{equation}\n\nDuring our tree searches, we solve millions of the above defined optimization problems on the fly with jDE~\\cite{brest2006self}, a self-adaptive differential evolution algorithm.\n\nThe GTOC~11 problem statement does not give any explicit constraints on the maximum $\\Delta V$ one mother ship can provide, but the objective function punishes excessive use of it. Thus, the typical trade-off between total transfer time and $\\Delta V$ needs to be taken into account. For the basic transfers, our focus is solely on minimizing $\\Delta V$ (i.e. the sum of all impulsive velocity changes over the three possible impulses) while keeping an upper bound (for our submitted solution of $380$ days) on $T$. As we show later on, our Lazy Race Tree Search is designed to assemble high ranking long chains of asteroid transfers and thus minimizes for the sum of all $T$ accordingly.\n\n\\subsection{Improving the Edelbaum Approximation}\n\\label{subsec:edelbaum_learn}\n\n\\begin{figure}[ht]\n\t\\centering\n     \\includegraphics[width=.9\\textwidth]{figures\/corrected_edelbaum.png}\n    \\caption{Error distribution across the test set for the Edelbaum approximation (average $\\approx 60$ days) and the learned corrected Edelbaum approximation (average error is  $\\approx 30$  days)}    \n    \\label{fig:corrected_edelbaum}     \n\\end{figure}\n\nEq.(\\ref{eq:arr_mass}) shows that the arrival mass of an asteroid depends largely on its transfer time $T$. A rough estimate $\\hat{T}$ of $T$ can be obtained by using an Edelbaum approximation for low-thrust time-optimal trajectories if we assume circular orbits~\\cite{kluever2011using}:\n\n\\begin{align}\n    \\label{eq:edeltree}\n    \\Delta V_{total} &= \\sqrt{V_{ast}^2 + V_{st}^2 - 2V_{ast}V_{st} \\cos{\\left[(\\pi\/2)\\Delta i\\right]}}\\\\\n    \\hat{T} &= \\frac{\\Delta V_{total}}{\\Gamma}\n\\end{align}\n\nwhere $V_{\\text{ast}} = \\sqrt{\\mu \/ a_{ast}}, V_{st} = \\sqrt{\\mu \/ a_{st}}$ are the circular velocities at departure and arrival, $\\Delta i = |i_{st} - i_{ast}|$ the desired inclination change and $\\Gamma$ the low-thrust acceleration.\nWhile $\\hat{T}$ is fast to compute, it neither accounts for the eccentricity nor the argument of perigee of the departure orbit, resulting in an error $T^* - \\hat{T}$ with regards to the optimal $T^*$ of the phase-free transfer as given by the solution of the corresponding OCP as given by Eq.(\\ref{eq:phasefree}). The fact that a sufficiently large population of such problems can be constructed and solved independently in parallel, allows us to apply supervised machine learning.\n\nIn particular, we are interested in regressing the error of the Edelbaum transfer time $\\hat{T}$:\n\\begin{equation}\n    T^* - \\hat{T} = f(a_{st}, a_{ast}, i_{ast}, e_{ast}, \\omega_{ast})\n\\end{equation}\nFor this purpose, we generated a database of $375\\,000$ solved optimal control problems by sampling $a_{st}$ uniformly at random within $[0.9, 1.4]$ AU and selecting asteroids randomly from all $83\\,453$ available candidates.\nThe values used to bound $a_{st}$ try to bracket as closely as possible our intuition on its optimal value and were not chosen following a rigorous approach. \n$250\\,000$ instances were used to train a multi-layer perceptron of five hidden layers of $50$ neurons with ReLU activation functions. \nThe remaining instances were used to evaluate the performance of the corrected Edelbaum approximation. \nFigure~\\ref{fig:corrected_edelbaum} shows the comparison between the uncorrected and our corrected Edelbaum approximation. As the decrease in error allows for more accurate predictions, we apply this improved approximator (denoted by $T_e$ in the following) to our tree search instead of the original Edelbaum estimation, reducing its error in almost all cases. \nThe impact on the run time of the tree search is negligible, as the inference of the neural network (once loaded into memory) is fast.\n\n\\subsection{Lazy Race Tree Search}\n\nThe Lazy Race Tree Search (LRTS) is a high level search strategy addressing combinatorial decisions for trajectory optimization problems. It is best understood as a variant of the widely known beam search~\\cite{bisiani1992beam, wilt2010comparison}, which itself is a generalization of the basic greedy search paradigm. In a greedy search, one would grow a chain of asteroids one by one by selecting only the best asteroid (according to some ranking criterion) at each step. However, for the asteroid selection problem, an optimal selection within each step does not imply optimality of the complete chain, i.e. Bellman's principle of optimality cannot be applied. Consequently, a greedy search will almost always be suboptimal.\n\nBeam search improves on the greedy search by considering not only the single best solution, but by keeping the $g$ best next asteroids saved inside a search tree structure, where $g$ is sometimes called ``branch-factor'' or ``fanout''. If one would allow this search tree to simply keep on branching, a chain of length $k$ would grow exponentially (i.e. $k^b$) which is why the beam search additionally limits the number of partial solution at each level of the search tree by the beam-width parameter $b$. In other words, each level of the tree prunes all intermediate solutions down to the $b$ best. This means that the search tree of a beam-search for a chain of $k$ asteroids requires at most $k \\cdot b \\cdot g$ evaluations, making it efficient to store and compute in practice while not being overly greedy.\n\nThe LRTS improves on the beam search concept by defining fair ranking criteria that lead themselves naturally to problems where ongoing time is an important aspect. It was developed and deployed successfully already during the GTOC 6, which required the assembly of a long multi-leg flyby tour around the Galilean moons~\\cite{Izzo2013Search}. Similarly to this challenge, our goal is the assembly of long asteroid chains with high arrival mass while maintaining a low $\\Delta V$. To achieve this goal, LRTS incrementally builds a search tree by expanding and pruning of nodes until the total mission time is exhausted. Each node in the search tree constitutes a chain of transfers beginning from Earth and visiting a number of asteroids $a_1, \\ldots, a_k$.\n\nThe specialty of LRTS is that it is not restricted to comparing solutions based on their chain length like a traditional beam search would do. Instead, nodes are ranked within a number of $s$ time-slices, beginning with the earliest time-slices (i.e. with the nodes that are ``slowest'' in the current race). The ranking of each node is computed by a variation of the objective function given in Eq.(\\ref{eq:objfun}) that ignores the bonus component and assumes ten identical mother ships:\n\n\\begin{equation}\n\\label{eq:lrts_ranking}    \nJ'(a_1, \\ldots, a_k) = \\frac{  M_e }{a^2_{D} \\cdot \\left(1 + \\Delta V_{k}^{Total} \/ 50 \\right)^2}\n\\end{equation}\nwhere $M_e$ is the sum of all expected arrival masses from all collected asteroids assuming a transfer time computed by the improved Edelbaum approximator. If the number of nodes within a time slice exceeds a certain beam width parameter $b$, the lowest ranking nodes are pruned. Thus, only the $b$ highest ranked nodes are expanded further into the future.\n\n\\begin{figure}[ht]\n\t\\centering\n     \\includegraphics[width=.9\\textwidth]{figures\/lrts.pdf}\n    \\caption{LRTS working on time-slice $\\Delta T_i$. The lower ranked nodes are pruned, the $b = 2$ highest ranked nodes are each expanded in $g = 3$ new nodes (bold and blue), sorted into different time-slices. The numbers inside the nodes correspond to the visited asteroid sequence. Note that LRTS ranks sequences of different length within the same time-slice.}    \n    \\label{fig:lrts}     \n\\end{figure}\n\nGiven a node with asteroid sequence $a_1, \\ldots, a_k$, the expansion (or branching) step of a node expands the sequence by another asteroid $a_{k+1}$. For that purpose, the final conditions of arrival at $a_k$ are considered and the orbital phasing indicator~\\cite{Izzo2016Designing} is computed to determine a distance from $a_k$ to all unvisited candidate asteroids. For a given fan-out parameter $g$, a kd-tree~\\cite{maneewongvatana1999s} is constructed to preselect the $2g$ highest ranking asteroids according to this indicator. For each preselected asteroid candidate, we then construct and solve an A2A-optimzation problem (see Eq.(\\ref{eq:a2a})) to determine the necessary $\\Delta V$ and transfer time (asteroid to asteroid) for this leg. Additionally, we use the improved Edelbaum approximator to estimate the transfer time $T_e$ of the asteroid to the ring station for a pre-defined semi-mayor axis $a_{D}$, resulting in an estimate of its arrival mass. This allows us to rank all $2g$ candidates according to their improvement on $J'$. Subsequently, only the best $g$ new nodes are inserted in their corresponding time-slice of the search tree and the rest is discarded.\nThus, expansion and pruning are both guided by $J'$, which takes all important factors of the original objective function into account. The lazy race aspect of ranking on time-slices assures a fair comparison which is unbiased by the length $k$ of the corresponding asteroids chains, which is not necessarily correlated with the objective function. Figure~\\ref{fig:lrts} shows conceptually how LRTS operates for one slice of time.\nLRTS terminates if there is no node in the search tree left that can be expanded, as the transfer time to the next asteroid would be outside the mission time. After termination, the highest ranking nodes according to $J'$ are extracted from the search tree, each of them describing a valid and complete mother ship trajectory. \n\n\\subsection{Mother Ship Ensemble Selection}\n\\label{subsec:mothership_ensemble_selection}\n\n\\begin{algorithm}[ht!]\n\\caption{Mother ship ensemble selection heuristic}\n\\label{alg:greedy}\n\\begin{algorithmic}\n\\Require $J_{init}, \\delta, \\mathcal{U} = \\{S_1, S_2, S_3, \\ldots \\}$\n\n\\State $J_{\\tau} \\gets J_{init}$\n\\State $\\mathcal{S} = \\emptyset$\n\\While{$|\\mathcal{S}| < 10$}\n    \\State $P \\gets \\{S \\in \\mathcal{U} \\,\\backslash\\, \\mathcal{S} ~|~ J'(S) > J_{\\tau}\\}$\n    \\State $S_R \\gets$ uniformly at random from $P$\n\\If{$S_i \\cap S_R = \\emptyset$ for all $S_i \\in \\mathcal{S}$}\n    \\State $\\mathcal{S} \\gets \\mathcal{S} \\cup \\{S_R\\}$\n\\ElsIf{$|\\bigcup\\limits_{S \\in \\mathcal{S}} S \\cap S_R| = 1$}\n    \\State $a \\gets \\bigcup\\limits_{S \\in \\mathcal{S}} S \\cap S_R$ \\Comment{$a$ is a single asteroid}\n    \\State $S_R' \\gets S_R \\,\\backslash\\, \\{ a \\}$\n    \\State $\\mathcal{S}' \\gets \\mathcal{S}$ removing $a$ from conflicting trajectory in $\\mathcal{S}$\n    \\If{$\\sum\\limits_{S \\in \\mathcal{S} \\cup \\{ S_R' \\}}\\frac{J'(S)}{|\\mathcal{S} + 1|} > J_{tr}$ or $\\sum\\limits_{S \\in \\mathcal{S'} \\cup \\{ S_R \\}}\\frac{J'(S)}{|\\mathcal{S} + 1|} > J_{tr}$}\n        \\If{$\\sum\\limits_{S \\in \\mathcal{S} \\cup \\{ S_R' \\}}\\frac{J'(S)}{|\\mathcal{S} + 1|} > \\sum\\limits_{S \\in \\mathcal{S'} \\cup \\{ S_R \\}}\\frac{J'(S)}{|\\mathcal{S} + 1|}$}\n            \\State $\\mathcal{S} \\gets \\mathcal{S} \\cup \\{S_R'\\}$\n        \\Else\n            \\State $\\mathcal{S} \\gets \\mathcal{S'} \\cup \\{S_R\\}$\n        \\EndIf\n    \\EndIf\n\\Else\n    \\State $J_{\\tau} \\gets J_{\\tau} - \\delta$  \\Comment{$\\delta$ is a small constant}\n\\EndIf\n\\EndWhile\n\\end{algorithmic}\n\\end{algorithm}\n\nConsidering a large amount of possible mother ship trajectories and their corresponding sequence of visited asteroid, the likelihood of having the same asteroid in two or more of these sequences is naturally increasing. If one would select two or more of such sequences visiting the same asteroid at different times, it would violate the validity of the solution according to the problem statement. In fact, the asteroid in question might not be even at the same position when the second visit occurs, as it could already have started moving towards the stations and thus altering its orbit. To circumvent this issue, our ensemble selection is built incrementally by only adding asteroid sequences that are pairwise disjoint to previously selected sequences or that can be fixed (by removing a double visited asteroid from one conflicting sequence) without sacrificing too much of (expected) $J$.\n\nFormalizing this idea, we model every single mother ship trajectory as a set $S_i = \\{a_1, \\ldots, a_k\\}$ of asteroids. The problem is now to find a collection of 10 such sets $\\mathcal{S} = \\{S_1, \\ldots, S_{10}\\}$, to which we refer to as mother ship ensemble, such that the expected $J(\\mathcal{S})$ is maximal while $\\mathcal{S}$ is pairwise disjoint, meaning $S_i \\cap S_j = \\emptyset$ for all $i,~j~\\in~\\{1,\\ldots,10\\}$. Assuming a large set $\\mathcal{U}$ of many possible trajectories $S_i$, this problem becomes quickly intractable, as it is closely related to the NP-complete maximum independent set problem in graphs. More precisely, if we model all $S_i$ as nodes in a graph and connect all $S_i, S_j$ for which $|S_i \\cap S_j| > 0$ holds, the problem is equivalent to finding a $10$-independent set with maximum $J$.\n\nConsequently, we decided to find our mother ship ensemble by a greedy heuristic, which incrementally builds $\\mathcal{S}$ by sampling and combining random $S_i \\in \\mathcal{U}$. Since our goal is to maximize the expected $J$ of the whole ensemble, we bias our sampling on a slowly decreasing threshold value $J_{\\tau}$ which limits possible candidates at the early phases of the algorithm to comparatively high $J'$ (oversampling). Furthermore, if the current ensemble $\\mathcal{S}$ and $S_i$ only overlap in one asteroid $a$, we check whether removing $a$ from either $S_i$ or the corresponding $S_j \\in \\mathcal{S}$ would be feasible, and if so, whether the average expected $J'$ from this fixed ensemble would be above $J_{\\tau}$, in which case we include $S_i$ into $\\mathcal{S}$ and proceed. Noticeably, this type of single-asteroid removal is a feature of our heuristic which goes beyond the above theoretical independent set problem formulation, but also requires some non-negligible time for reoptimization of trajectories. Algorithm~\\ref{alg:greedy} describes our heuristic in pseudocode.\n\n\n\\section{Ring Building Pipeline}\n\nFollowing the construction of ten mother ship trajectories (a mother ship ensemble) in the previous section, the starting point of our ring building pipeline is a corresponding set of $N_{tot}$ asteroids $\\mathcal{A} = \\bigcup\\limits_{S \\in \\mathcal{S}} S$ and their earliest possible activation epochs $t^{act}_{i}, i = 1 .. N_{tot}$. The pipeline proceeds through three steps: the computation of optimal ring parameters, the optimization of the phased asteroid transfers and the scheduling of their arrival to the target stations. The results of these steps are, correspondingly, the exact orbits of the twelve Dyson ring stations, optimally-phased transfer trajectories for each asteroids and a set of construction time windows for each of the twelve stations, resulting in a complete solution for the GTOC~11 challenge.\n\n\\subsection{Computing the Ring Parameters}\n\\label{subsec:ring_parameters}\n\nThe competition requires the orbit of the Dyson ring to be circular $e_D = 0$, but leaves its semi-major axis (and radius) $a_D$, inclination $i_D$ and right ascension of the ascending node $\\Omega_D$ free (as well as a starting station phase $\\varphi_D$ which we decided to set and keep to zero). Following the objective function Eq.(\\ref{eq:objfun}), our design goal is to select these free parameters to allow favorable (i.e. short and well-phased) transfers for all activated asteroids to increase $M_{min}$ while also reducing $a_D$ as much as possible. A precise computation of $M_{min}$ (and thus a direct optimization for $J$) is prohibitive at this point, as we would need to know the exact transfer times for each asteroid and a (potentially optimized) schedule for each station. However, this information becomes only accessible to us at the end of the pipeline and not at its beginning. Consequently, we define an approximate objective function $\\bar{J}$ under the following simplifying assumptions:\n\n\\begin{itemize}\n    \\item Instead of the actual phased transfer times of each asteroid in $\\mathcal{A}$ to any given station of the Dyson ring, we use the corrected Edelbaum approximation described in Section~\\ref{subsec:edelbaum_learn} as a phase-less substitute, denoted by $T_{e,i}$ in the following. \n    \\item Only asteroids in $\\mathcal{A}$ capable to deliver a non-zero arrival mass to any station before the final epoch $t_f$ are relevant for the ring construction. Thus, we define a subset $\\mathcal{A}_e \\subset \\mathcal{A}$ of relevant asteroids as followed:\n\\[\n    \\mathcal{A}_e = \\lbrace i \\in \\mathcal{A} \\mid t^{act}_{i} + T_{e,i} < t_f \\text{ and } m_{0,i} \\cdot (1 - \\alpha \\cdot T_{e,i}) > 0 \\rbrace.\n\\]\n    \\item Although $\\mathcal{A}_e$ is not necessarily equal to $\\mathcal{A}$, we still assume the complete $\\Delta V$ required for visiting all asteroids $\\mathcal{A}$. Thus, the potential decrease in $\\Delta V$ by re-optimizing the mother ship trajectories to only visit the relevant asteroids $\\mathcal{A}_e$ is not accounted for at this point.\n    \\item Imbalances in the mass-distribution of the stations are also not accounted for. Instead, we assume that the maximum deliverable mass\n\\[\n     M_{tot} = \\sum\\limits_{i \\in \\mathcal{A}_e}{}m_{0,i} (1- \\alpha \\cdot T_{e,i}) \n\\]    \n    is achieved and equally distributed so that each station receives a mass of $M_{tot} \/ 12$.\n\\end{itemize}\n\nGiven above assumptions, we define the following optimization problem:\n\n\\begin{equation}\n\\label{eq:ringparameters} \n    \\begin{array}{rl}\n        \\mbox{find:} & a_D, i_D, \\Omega_D\\\\\n        \\mbox{to maximize:} & \\bar{J} = \\dfrac{\\sum\\limits_{i \\in \\mathcal{A}_e}{}m_{0,i} (1- \\alpha \\cdot T_{e,i})}{a^2_{D} \\sum\\limits_{k=1}^{10}\\left(1 + \\Delta V_{k}^{Total} \/ 50 \\right)^2}\\\\\n        \\mbox{subject to:}&  0.65 \\leq a_D \\leq 5 \\\\\n        &  0 \\leq i_D \\leq \\pi \\\\\n        & 0 \\leq \\Omega_D \\leq 2\\pi\n    \\end{array}\n\\end{equation}\n\nDuring the competition, we constructed and solved multiple of these optimization problems by Co-variance Matrix Evolutionary Strategy (CMA-ES)~\\cite{hansen2001completely}, allowing us to fix the orbital parameters of the ring for the subsequent steps of our pipeline.\n\n\\subsection{Optimally-phased Trajectories}\n\\label{subsec:optimally_phased_traj}\nGiven the orbital parameters $a_D, i_D$ and $\\Omega_D$ of the Dyson ring stations from the previous pipeline step, we now seek to find efficient transfer opportunities for all asteroids $\\mathcal{A}$ to all of the twelve target stations. These are trajectories that reach a station time-optimally when the arrival epoch is left free. In this sense, we refer to them as optimally-phased trajectories. Every asteroid should preferably travel along one of these trajectories to its assigned station within the corresponding station building window, as this will ensure an efficient use of its mass. Since the station building windows will be assigned later, we assume for each asteroid $i \\in \\mathcal{A}$ an arrival interval of $[t_i^{act}, t_f]$, where $t_f$ denotes the end of the whole mission. Thus, given this interval, the task is to find all the local minima of the optimization problem stated in Eq.(\\ref{eq:rdvz}) with respect to each of the twelve target stations. \n\n\\begin{figure}[ht]\n\t\\centering\n     \\includegraphics[width=.9\\textwidth]{figures\/phase.png}\n    \\caption{Each dot represents an optimally-phased transfer opportunity (marking its arrival time) for an example asteroid at $a=2.32$, $e=0.031$ and $i=1.14\\degree$, earliest possible activation time $5.3$ years after mission start. The blue vertical lines mark the end of the mission and the earliest possible transfer opportunity within the mission time frame. The Edelbaum approximated transfer time for the particular asteroid shown is $T_{e,i} = 982.5$ days, while the actual minimum transfer time is $T_{min,i} = 990$ days and the highest is $T_{max,i} = 1088$ days. A brighter color indicates a qualitatively higher arrival mass at the corresponding station.}\n    \\label{fig:well_phased}     \n\\end{figure}\n\n\nWe will formally collect these transfers in a matrix $\\mathbf M$, whose $i,j$ entry is a set $(t_k, m_k), k=1..K_{ij}$ containing all $K_{ij}$ epochs $t_k$ at which the mass $m_k$ from asteroid $i$ can be brought to the station $j$ using an optimally-phased transfer. To compute $\\mathbf M$ efficiently, we make use of the following trick: In a first step, we find for asteroid $i$ a single optimally phased transfer to station $j=1$. This is done by restricting its arrival time window $[\\underline t_f, \\overline t_f]$ to the interval $[t_i^{act}, t_i^{act} + \\delta T_{e,i}]$, using the corrected Edelbaum approximation $T_{e,i}$ presented in Section~\\ref{subsec:edelbaum_learn} and some slack $\\delta > 1$. We denote the time of this transfer by $T_i$. After determining $T_i$ by solving this OCP instance, all other transfers can be found by solving similar OCPs, which can now be constructed by exploiting an apparent periodicity in the solutions space. This periodicity is visualized for an example asteroid in Figure~\\ref{fig:well_phased}. While the periodic structure depends on the orbital elements of each asteroid and becomes less linear and predictable for higher inclinations or eccentricities, it nevertheless is strong enough to infer excellent initial guesses for the solvers to use.\n\nThe initial guesses are constructed by computing the synodic period $T_{syn,i}$ between the orbit of asteroid $i$ and the Dyson ring orbit. Given the actual transfer time $T_i$ as solution from the first OCP, all other solutions from asteroid $i$ to the first station $j = 1$ can be found by adding multiples of $T_{syn,i}$ and bracketing the arrival window $[\\underline t_f, \\overline t_f]$ to narrow time intervals around these points:\n\\[\n[t_i^{act} + T_i + k \\cdot T_{syn,i} - \\varepsilon, t_i^{act} + T_i + k \\cdot T_{syn,i} + \\varepsilon]\n\\]\nfor all $k$ that allow us to remain within the mission time. To find the transfers from $i$ to all other stations, we followed a similar procedure, by adding an additional offset of $j \\cdot \\frac{T_{syn,i}}{12}$ for every station $j = 2, \\ldots, 12$ to the bracketing of the arrival window. \n\n\n\n\n\n\\subsection{Ring Building Schedule}\n\\label{subsec:ring_buidling_schedule}\n\nFollowing the preceding generation of $\\mathbf M$, it remains to select for each asteroid, which of the many possible transfer opportunities it ultimately should take. This problem is closely related to the classical combinatorial assignment problem~\\cite{burkard2012assignment}, where asteroids can be thought of as agents and stations as tasks. However, additional complexities arise in our case: the largest constraint is that only a single station can receive asteroids at a time. Formally, we define a set of receiving time windows $\\mathcal{W} = \\lbrace (W_\\mathrm{1,begin}, W_\\mathrm{1,end}), \\ldots, (W_\\mathrm{12,begin}, W_\\mathrm{12,end}) \\rbrace$ for each of the twelve stations, where for one station $j$ the earliest and the latest epoch at which it accepts incoming asteroids is described by $W_{j,\\mathrm{begin}}$ and $W_{j,\\mathrm{end}}$ correspondingly. Furthermore, we require that there is no overlap between any pair of time windows and that a gap of at least 90 days is left between temporally consecutive windows, as demanded by the problem formulation of the challenge.\n\nGiven that the $\\Delta V_k^{total}$ and $a_D$ component of $J$ (overall objective function, see Eq.(\\ref{eq:objfun})) are fixed at this point during the pipeline, the task of the ring building schedule remains to solve the assignment problem and determine a valid set of time windows $\\mathcal{W}$ such that the last important component $M_{min}$ is maximized. For this purpose, we model the scheduling task as a bi-level optimization problem: The outer level problem encodes the intervals of a candidate window allocation $\\mathcal{W}$ while the inner level solves the corresponding assignment problem using $\\mathbf M$ constrained by $\\mathcal{W}$.\n\nFor the outer level, we deploy a simple differential evolution solver to find optimal $\\mathcal{W}$, guided by $M_{min}$ (the solution of the inner optimization problem) as an objective. For the inner level problem we deploy a modified Hungarian algorithm to find a solution to the specific instances of the constrained assignment problems. In the following, we describe the necessary steps of the latter procedure in detail.\n\n\\subsection{Inner Level Problem: Path-Based Refinement}\n\\label{subsec:path_based_refinement}\nGiven the matrix $\\mathbf M$ and a time window allocation $\\mathcal{W}$, we create a filtered matrix $\\mathbf M'$ by only allowing transfers from $\\mathbf M$ that arrive at their destination within the time windows defined by $\\mathcal{W}$. Thus, an entry in $\\mathbf{M'}_{ij}$ might be empty if for asteroid $i$ no arrival to station $j$ within $(W_{j,\\mathrm{begin}}, W_{j,\\mathrm{end}})$ is available. Should there be multiple arrivals available, we select the transfer that allows for a maximum mass to be delivered. More formally, $\\mathbf{M'}_{ij} = \\mathbf{M}_{ijk}$, with \n\n\\begin{equation}\nk = \\argmax_{m_k}  ((t_k, m_k) \\in \\mathbf{M}_{ij} \\mid W_{j,\\mathrm{begin}} \\leq t_k \\leq W_{j,\\mathrm{end}})\n\\end{equation}\n\nConsequently, for each asteroid-station pair, at most one transfer opportunity is left in $\\mathbf{M'}_{ij}$. The problem remains to decide however, which asteroid will go to which station, i.e. which of the asteroid-station pairs to select without violating the constraint that one asteroid may be only assigned to one station.\n\nWe model this problem as a weighted bipartite graph, with all asteroids on one side, all stations on the other side and an edge between an asteroid $i$ and a station $j$ if $i$ can reach $j$ with non-zero mass. The weight of the edge $ij$ is the mass that asteroid $i$ can deliver to $j$, as given by $\\mathbf{M'}_{ij}$.\n\nFinding an optimal asteroid-to-station assignment is then equivalent to finding a matching in this graph which maximizes the mass arriving at the minimum mass station. Figure~\\ref{fig:hungarian-sketch-initial} illustrates such an assignment graph for a simple example of six asteroids and three stations, assuming equal arrival masses for each transfer.\n\n\\begin{figure}\n\t\\centering\n    \\subcaptionbox{\\label{fig:hungarian-sketch-initial}}{\\includegraphics[width=0.2\\textwidth]{figures\/hungarian-sketch.png}}\n    \\subcaptionbox{\\label{fig:hungarian-sketch-path}}{\\includegraphics[width=0.2\\textwidth]{figures\/hungarian-sketch-path.png}}\n    \\subcaptionbox{\\label{fig:hungarian-sketch-optimized}}{\\includegraphics[width=0.2\\textwidth]{figures\/hungarian-sketch-optimized.png}}\n    \\caption{Bipartite graph of a simple asteroid to station assignment problem. Six asteroids are assigned to three stations, each edge corresponds to a possible transfer. Solid lines signify a matched edge (i.e., a selected transfer), dashed lines an unmatched edge. Figure~\\ref{fig:hungarian-sketch-initial} shows an unbalanced assignment: Station 2 has only one asteroid, and asteroid 1 is free but cannot be assigned to it, as it has no possible transfer. Figure~\\ref{fig:hungarian-sketch-path} shows an augmenting path, drawn in thick blue lines, alternating matched and unmatched edges. Figure~\\ref{fig:hungarian-sketch-optimized} shows the improved solution: Switching matched and unmatched edges on the augmenting path.}\n    \\label{fig:hungarian-sketch}\n\\end{figure}\n\nWhile for a classical assignment problem the sum of the weights of the matching is to be maximized, our objective is to maximize the minimum total weight over all stations. This difference makes the problem NP-hard, as the multiway number partitioning problem~\\cite{doi:10.1137\/0117039}, specifically maximizing the smallest sum, can be reduced in polynomial time to it.\\footnote{Sketch of proof: Given a $k$-way number partitioning problem with a multiset $S$ of numbers, create $k$ building stations and one asteroid for each number $s_i \\in S$, with a mass equal to $s_i$. For each value $m \\in \\mathbb{R}$, there is a partition of $S$ into $k$ subsets with sum at least $m$ exactly if there is an allocation of asteroids with the minimum-mass station having a mass of at least $m$.}\n\nConsequently, finding an exact solution for large instances of the asteroid assignment problem is prohibitive. Thus, we deploy a heuristic inspired by the Hungarian algorithm~\\cite{kuhn1955hungarian} for the assignment problem.\n\nGiven an assignment graph $G$, we define an alternating path in $G$ to be a connected sequence of edges, alternating between matched and unmatched edges. Figure~\\ref{fig:hungarian-sketch-path} shows an alternating path of length $k=4$: station 2 $\\leftrightarrow$ asteroid 2, $\\leftrightarrow$ station 1 $\\leftrightarrow$ asteroid 1. Switching all edges from assigned to unassigned and vice versa yields another valid assignment graph, shown in Figure~\\ref{fig:hungarian-sketch-optimized}. We define the gain of a path with respect to the cost as the difference in value caused by applying it. Our approach for improving our assignment consists of repeatedly finding and applying augmenting paths with positive gain, as described in Algorithm~\\ref{alg:hungarian}. \n\n\\begin{algorithm}[ht!]\n\\caption{Path-based Refinement of asteroid-station allocation}\n\\label{alg:hungarian}\n\\begin{algorithmic}\n\\Require{transfer matrix  $\\mathbf M$, time windows $\\mathcal{W}$, maximum path length $k$, [initial assignment]}\n\\State $\\mathbf{M'} \\leftarrow$ filter transfers in $\\mathbf M$ by windows $\\mathcal{W}$\n\\State $G \\leftarrow$ create assignment graph from $\\mathbf{M'}$, setting all edges to unmatched\n    \\For{asteroid-station pair $(i, j)$ in initial assignment}\n\t\\State mark edge in $G$ as matched\n\t\\EndFor\n\\While{improvement found} \\Comment{Main Loop}\n\t\\For{station $s$} \\Comment{Sorted by increasing mass}\n\t\t\\State paths $\\leftarrow$ generate alternating paths of length $k$, starting from $s$\n\t\t\\State gains $\\leftarrow$ calculate gains of all paths\n\t\\EndFor\n\t\\If{maximum gain $>$ 0}\n\t\\State apply path of maximum gain to $G$\n\t\\EndIf\n\\EndWhile\n\\State \\Return assignment from $G$\n\\end{algorithmic}\n\\end{algorithm}\n\nRegarding this algorithm, there a few things to note: First, since the number of paths of length $k$ grows exponentially, the running time is exponential in $k$ as well. In practice, we observed that the increase in running time for any $k>4$ does not justify the improvements in our objective function. We thus limit the path length $k$ for this step of our pipeline to $k=2$ for use in the differential evolution, and $k=4$ for post-processing refinements\nSecondly, the algorithm benefits from being bootstrapped by an initial assignment. We construct this initial assignment by deploying a greedy scheduling algorithm, described in Algorithm~\\ref{alg:greedy_allocation}. Given an initial target mass $m$, the greedy scheduling incrementally increases the mass of all yet unconstructed stations by always selecting the earliest possible arrival opportunity. Once a station $j$ accumulates a mass larger than $m$, it is considered constructed and we can infer a construction time window $(W_{j,\\mathrm{begin}}, W_{j,\\mathrm{end}})$ by its earliest and latest arriving asteroid. Once a station $j$ is constructed, all asteroids arriving in its time window are fixed to the station and the accumulated masses in the remaining stations are set again to zero as only one station can be build at a time. After adding the gap of $90$ days to $W_{j,\\mathrm{end}}$, we proceed to fill up the remaining stations using the earliest arrival of the yet unassigned asteroids. The algorithm terminates once all twelve stations are successfully constructed (in which case the minimum mass station has a mass of at least $m$) or if no more assignments can be added, potentially leaving stations unconstructed or with low accumulated mass. To avoid the latter outcome, the target mass $m$ needs to be set conservatively, i.e. by increasing $m$ in small amounts and restarting the greedy scheduler until the best greedy assignment is found.\n\n\\begin{algorithm}\n\\caption{Greedy scheduling}\n\\label{alg:greedy_allocation}\n\\begin{algorithmic}\n\\Require{transfer matrix $\\mathbf M$, target average mass $m$}\n\\For{$(t_k, m_k) = \\mathbf{M}_{ijk}$ chronologically}\n\t\\State add $m_k$ to $mass(j)$\n\t\\State add asteroid $i$ to $A(j)$\n\t\\State advance time to $t_k$\n\t\\If{unconstructed station $j$ with $mass(j) > m$ exists}\n\t\t\\State fix asteroids $A(j)$ and construct station $j$\n\t\t\\State determine $(W_{j,\\mathrm{begin}}, W_{j,\\mathrm{end}})$ by earliest and latest arriving asteroid\n\t\t\\State reset $A(l) = \\emptyset$ for all stations $l$ that still need construction\n\t\t\\State reset $mass(l) = 0$ for all stations $l$ that still need construction\n\t\t\\State exclude station $j$ and all asteroids $i \\in A(j)$ from $\\mathbf M$\n\t\t\\State advance time to $W_{j,\\mathrm{end}} + 90$d\n\t\\EndIf\n\\EndFor\n\\State \\Return assigned asteroids $A(j), j = 1, \\ldots, 12$ and time windows $\\mathcal{W}$\n\\end{algorithmic}\n\\end{algorithm}\n\n\\section{Results}\n\n\\subsection{Mother Ship Tree Searches}\n\nDue to the time limitations of GTOC~11 we manually balanced the various hyper-parameters of our LRTS during experiments in order to quickly generate a large collection of feasible single mother ship trajectories of high $J'$. The most common choices for beam-width were $b \\in \\{10,20,30\\}$, for fanout $g = \\{5, 10, 20\\}$ and $a_{D} \\in \\{1.2, 1.25, 1.3, 1.35, 1.40\\}$ AU. The time of flight for each A2A-leg was bounded by $380$d and for the initial E2A-leg by $480$d. Out of all available asteroids, we computed the arrival mass $m_{arr}(a_D)$ for a fixed $a_{D}$ using the improved Edelbaum approximation and considered only asteroids above the $0.9$-quantile as valid targets. The length of a time-slice was fixed to $91.3125$d, which amounts to a total of $80$ time-slices for the whole mission duration. Executing the tree search in parallel on our servers and a high performance computation cluster allowed us to generate $145\\,000$ raw single mother ship solutions, of which we selected the $7\\,000$ best (according to $J'$) for further optimization.\n\n\\begin{figure}[h!]\n\t\\centering\n     \\includegraphics[width=\\textwidth]{figures\/mothers_overview.png}\n    \\caption{Ensemble of ten mother ship trajectories. The vertical lines correspond to asteroid activation epochs. The blue circles are plotted in correspondence to the $\\Delta V$ manoeuvres epochs and have size proportional to the magnitude. The red dots mark the cumulative asteroid mass accumulated in units of the maximal mass $m_{max}$ of the dataset.} \\label{fig:ensemble}     \n\\end{figure}\n\nGiven our approach of constructing the mother ship trajectories leg by leg using the simplified three-impulse model (see Section~\\ref{subsec:a2a}), the $\\Delta V$ requirements still had potential for improvement by local optimization starting from the full asteroid sequence (without altering it). For that purpose, we deployed the sequential least squares programming (SLSQP) solver from Python's scipy library for 10 iterations on each trajectory. Obtaining lower $\\Delta V$ resulted in a new ordering of the $7\\,000$ trajectories according to $J'$, which constitutes the set $\\mathcal{U}$ for the following mother ship ensemble selection. Due to its randomness, we executed Algorithm~\\ref{alg:greedy} multiple times to generate promising mother ship ensembles. Figure~\\ref{fig:ensemble} shows the final ensemble, which was part of our highest scoring submission. In total, this ensemble collects $301$ asteroids with a DV-factor $\\sum\\limits_{k=1}^{10}\\left(1 + \\Delta V_{k}^{Total} \/ 50 \\right)^2 \\approx 19.1269$. \n\n\\subsection{Ring Building Pipeline}\n\nFollowing the steps of the pipeline, we determined a station ring radius of $a_D = 1.3198$ AU, an inclination of $i_D = 1.4282\\degree$ and a right ascension of the ascending node $\\Omega_D = 104.4039\\degree$ by solving the optimization problem outlined in Eq.(\\ref{eq:ringparameters}) using Co-variance Matrix Evolutionary Strategy (CMA-ES). The phase $\\phi_1$ of the first station was not optimized and set to $0\\degree$, resulting in a general phase of $\\phi_j = 30\\degree \\cdot (j - 1)$ for all stations $j = 1, \\ldots, 12$. \n\nIn a next step, we constructed for each asteroid of the mother ship ensemble a first transfer to the first station $j=1$ as described in Section~\\ref{subsec:optimally_phased_traj}. The resulting OCP (compare Eq.(\\ref{eq:rdvz})) is solved by randomly initializing the Position and velocity co-states in $[-1, 1]$ and deploying SNOPT, which resulted in $301$ initial transfer solutions. Out of those, we construct the complete matrix $\\mathbf M$ of all possible optimally-phased transfer opportunities for each asteroid to each station. Following this procedure, additional 14\\,955 OCPs were generated and solved in parallel on our server equipped with 2xAMD EPYC ROME 64-CORE 7702 3.35GHZ CPUs, which can solve this amount of OCPs in less than one hour. The resulting transfer matrix $\\mathbf M$ contained $15\\,256$ transfer opportunities, approximately $4$ for each asteroid-station pair (more or less depending on the activation epoch and orbit of the asteroid in question).\n \nFollowing the construction of the matrix $\\mathbf M$, the greedy algorithm described in Algorithm~\\ref{alg:greedy_allocation} was used to bootstrap the ring schedule optimization. Given $m_{max} = 1.963 \\cdot 10^{14}$ kg to be the highest mass among all asteroids of the problem description, we assigned a starting mass of $m = 9m_{max}$ to the greedy algorithm and increased $m$ by steps of $\\Delta m = 0.05m_{max}$. The highest $m$ for which the greedy scheduler still found an equally distributed balance was $m = 9.5m_{max}$. Given the assignment from the greedy scheduler and its time window allocation $\\mathcal{W}$, the bi-level optimization problem as described in Section~\\ref{subsec:ring_buidling_schedule} was constructed. The outer optimization problem was solved deploying an asychronous island model with three islands, each with a population of size $10$ initialized by $\\mathcal{W}$. The modified Hungarian algorithm (Algorithm~\\ref{alg:hungarian}) was restricted to paths of length $k=2$ for solving the inner optimization problems, returning $M_{min}$ as fitness. Thus, for the outer optimization problem, a differential evolution algorithm modified $\\mathcal{W}$ by selecting the individuals of high $M_{min}$ in the population for recombination and propagation to the next evolutionary generation.\n\nSince this fitness function is costly to evaluate (as it involves solving the asteroid assignment problem) we could run differential evolution for ten generations only, which nevertheless was sufficient to improve $M_{min}$. After this step, we applied some post-processing refinements to the best found solution of the assignment problem by running Algorithm~\\ref{alg:hungarian} with path lengths of $k=3$ and $k=4$. Overall, the initial minimum station mass of $m = 9.5m_{max}$ constructed by the greedy scheduler could be improved to $m =  10.24m_{max}$ which is our final $M_{min}$. \n\n\n\\begin{table}[h]\n\\centering\n\\begin{tabular}{|c|c|}\n\\hline\n$a_D$ [AU]                  & $1.3198$                 \\\\ \\hline\n$i_D$ [$\\deg$]              & $1.4282$                 \\\\ \\hline\n$\\Omega_D$ [$\\deg$]         & $104.4039$               \\\\ \\hline\n$M_{min}$ [kg]              & $2.0125 \\cdot 10^{15}$   \\\\ \\hline\n$N_{tot}$                   & $301$                    \\\\ \\hline\n$J$ [-]                     & $6359.7249$              \\\\ \\hline\n\\end{tabular}\n\\caption{Characteristics of the ACT\\&Friends' final submitted solution to the GTOC~11 challenge, ranked second place in GTOC~11.}\n\\label{tab:finalsolution}\n\\end{table}\n\nEverything taken together, the parameters of the solution that our pipeline produced are summarized in Table~\\ref{tab:finalsolution}. Our server finished this computation in about 2-3 hours with the majority of time spent on OCP solving and the optimization loops for the scheduling problem. The computational cost depends largely on the parallelization capabilities of the hardware and is thus not easily quantifiable. Our overall score of $J = 6359.7249$ (not accounting for the bonus factor $B$) allowed us to place second in GTOC~11. A short animation showcasing the trajectories for the construction of our Dyson ring can be found online~\\cite{ourvideo} alongside the winning solution of Tsinghua university.\n\n\\section{Conclusions}\n\nThe presented work of this report was developed with the goal in mind to obtain the highest possible rank during GTOC~11. Given the fiercely competitive nature of recent GTOCs, our team was forced to innovate on our techniques, which would not have been sufficient by themselves. One innovation was the addition of machine learning to improve accuracy and provide better informed decisions.\n\nAs numerous OCP problems are typically encountered during GTOCs but also for complex trajectory optimization tasks in general, it is of importance to utilize optimal solutions not only directly (i.e. to advance a combinatorial search) but also indirectly, by building databases for machine learning. Distilling a model to correct the error of the Edelbaum approximation for example enabled mayor improvements for our tree search algorithms, but was also essential for multiple independent steps of our ring building pipeline, due to the strong relation between $T_{e}$ and $M_{min}$.\nConsidering the difficult fitness landscapes of the OCP problems involved, rapid deployment of fast evolutionary optimization techniques are still essential for success. While linking small building blocks of evolutionary optimization is standard practice to solve combinatorial problems via tree searches or ant colony optimization, an additional innovation described in this report was the application of the modified Hungarian algorithm as part of an evolutionary outer loop. This provides a fusion of traditional assignment\/scheduling algorithms with techniques from evolutionary optimization which turned out to be highly effective. \n\nAlthough we might still need a few more centuries to see something like the Dyson ring become a reality in our solar system, the current trends regarding infrastructure in space are already visible. Consequently, one may safely assume that scheduling and planning problems similar to the GTOC~11 challenge will only increase in relevance and thus provide a new and worthwhile opportunity to study trajectory optimization at larger scales.\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
{"text":"\\section{Introduction}\n\nThere are numerous effects in physics which are dependent on the\ndimension of the spacetime in which they live, and concepts which\nhave mathematically simple models when restricted to one or two\ndimensions may become intractably complex as the dimensionality\nincreases. Furthermore, in some situations there is a sharp contrast\nin the behaviour of the system when the number of dimensions passes\nabove or below a certain ``critical dimension''. In the field of\ngeneral relativity there are already some instances of such\ndimension dependent phenomena; for example, the Gregory-Laflamme\ninstability \\cite{greglaf1,greglaf2} of black strings, and the work\nof Belinsky, Khalatnikov and Lifshitz (BKL) \\cite{bkl1} and its\nextensions \\cite{dema1,dema2,elsk}, where the dynamics of a\nspacetime in the vicinity of a cosmological singularity were\nstudied. In the latter case they found that the general behaviour of\nthe relevant Einstein solutions changed from ``chaotic'' in the low\ndimensional cases to non-chaotic in higher dimensions. It is this\nextra complexity and appearance of a critical dimension which is\ndiscovered in the work presented here.\n\nThe issue of dimensionality has become of even greater importance\nover the past few decades, with the theories of hidden dimensions\nfirst postulated by Kaluza \\cite{kaluza} resurfacing in the quest\nfor unification and a Theory of Everything. Originally it was hoped\none extra dimension would be sufficient; with the advent of string\ntheory and its subsequent developments this was then expanded to\ntwenty six in the late 1960's as a consistency requirement for\nbosonic strings, before being reduced back down to ten with the\nintroduction of supersymmetry in the 1980's. The idea of holography\nalso has its roots in dimensionality, and its most famous form, that\nof Maldacena \\cite{adscft}, states a relation between string theory\non a five-dimensional anti-de Sitter space, and a four-dimensional\nconformal field theory. Much work (see \\cite{aha} for a review) has\nfollowed investigating spacetimes and conformal field theories of\nvarying dimension and complexity.\n\nHere we present an analysis into the stability of radiating perfect\nfluid spheres in an asymptotically anti-de Sitter spacetime. Due to\nthe confining nature of the AdS potential, the spheres of radiation\nare self-gravitating, and thus we shall refer to them as ``stars''\nin much of the following, although this is mainly used as a\nconveniently brief label, as it is only a toy-model approximation to\na star at best.\n\nWe begin by giving the equations for a such a model in\n$d$-dimensions, before analysing the behaviour of the star's total\nmass. We consider the variation of the total mass as a function of\nthe central density, and observe that for large enough\ndimensionality, the mass increases monotonically with the density.\nHowever in the lower dimensional cases, oscillations appear (this\nwas originally noted in the $d=5$ case in \\cite{hubenynew}),\nindicating that the perfect fluid model of the star is becoming\nunrealistic. We numerically find the critical dimension separating\nthese two regimes to be $11.0$ (to three significant figures), and\ngive an explicit relation, \\eqref{eq:ch6peakest1}, between the\nspacetime dimension $d$ and the ``saturation density'' $\\rho_{c}$,\nsee section \\ref{sec:criticald}. The existence of local maxima at\ncritical central densities (saturation points) in the lower\ndimensional cases indicates the appearance of instabilities in the\nmodel of the star, and point to unrealistic effects developing for\n$\\rho_{0} > \\rho_{c}$. We also provide a numerical analysis of the\nbehaviour at large central density, in particular the self-similar\nbehaviour that appears in dimension $d < d_{c}$; several parameters\nof our numerical model are then also determined analytically from a\ndynamical systems analysis of the behaviour, where we consider the\nexpansion about a fixed point of the zero-cosmological constant\nsolution.\\footnote{This dynamical systems analysis (section\n\\ref{sec:dynamsys}) was suggested after correspondence with V.~\nVaganov, who also considered the behaviour of self-gravitating\nradiation in $AdS_{d}$ in \\cite{vaga}. Work simultaneously conducted\nby P.~H.~Chavanis also found the critical dimension described here,\nvia an alternative route, in his comprehensive study of relativistic\nstars with a linear equation of state, \\cite{chan} (see the note\nfollowing the discussion for more details).}\n\nThe outline of the paper is as follows: we begin with a brief recap\non perfect fluid models in general dimension in section\n\\ref{sec:fluidmodels}. In section \\ref{sec:analysis} we introduce\nthe total mass as a function of the central density, analyse the\nprogression of the saturation point with increasing dimension, and\npresent the best fit formula, \\eqref{eq:ch6peakest1}, which gives a\nvalue for the critical dimension. We also present further numerical\nresults for the behaviour of the total mass at large central\ndensities. In section \\ref{sec:dynamsys} we give a dynamical systems\nanalysis, following the methods in \\cite{vaga} and \\cite{uggla},\nwhich yields analytical expressions for several of the numerical\nresults of the previous section. Finally, we conclude with a\ndiscussion of the results in section \\ref{sec:discussion}.\n\n\\section{Perfect fluid models} \\label{sec:fluidmodels}\n\nConsider a general static, spherically symmetric $d$-dimensional AdS\nspacetime with metric:\n\\begin{equation} \\label{eq:ch44AdSmetric}\nds^{2} = - k(r) dt^{2} + h(r) dr^{2} + r^{2}d \\Omega_{d-2}^{2}\n\\end{equation}\nBy considering a perfect fluid of a gas of radiation, one can obtain\nimplicit expressions for $k(r)$ and $h(r)$ for a simple model of a\n``star'' geometry. For a perfect fluid we have that the stress\ntensor is of the form:\n\\begin{equation} \\label{eq:ch44stresstensor}\nT_{ab} = \\rho(r) u_{a} u_{b} + P(r) (g_{a b} + u_{a} u_{b})\n\\end{equation}\nwhere $u^{a}$ is the $d$-velocity of a co-moving gas, and upon which\nwe impose the further constraint that the matter be purely\nradiating; this sets $\\rho(r) = (d - 1) P(r)$ as it requires that\n$T_{ab}$ be traceless. One obtains the required metric by solving\nEinstein's equation: $G_{a b} + \\Lambda g_{a b} = 8 \\pi G_{d} T_{a\nb}$, with the above stress tensor and a negative cosmological\nconstant, as follows. The relevant components of Einstein's\nequations in general dimension $d$ are given by:\n\n\\begin{equation} \\label{eq:ch44Grr}\nG_{r r} = \\left(\\frac{d-2}{2}\\right) \\frac{(d-3)(k(r) - k(r) h(r)) +\nr k'(r)}{r^{2} \\, k(r)} = \\left(\\frac{\\rho(r)}{d-1} + \\frac{(d-1)\n(d-2)}{2 \\, R^{2}} \\right) h(r)\n\\end{equation}\n\\begin{equation} \\label{eq:ch44Gtt}\nG_{t t} = k(r) \\left(\\frac{d-2}{2}\\right) \\frac{(d-3)(h^{2}(r) -\nh(r)) + r h'(r)}{r^{2} \\, h^{2}(r)} = \\left(\\rho(r) + \\frac{(d-1)\n(d-2)}{2 \\, R^{2}} \\right) k(r)\n\\end{equation}\nwhere we have used that $\\Lambda = - (d-1)(d-2)\/(2 R^{2})$, $P(r) =\n\\rho(r)\/(d-1)$, and set $8 \\pi G_{d} \\equiv 1$ for\nconvenience\\footnote{In the numerical results presented shortly we\nalso set $R = 1$; we include it here to ease comparison with the\ndynamical systems analysis of the $\\Lambda = 0$ ($R = \\infty$) case\ngiven in section \\ref{sec:dynamsys}.}. We can infer the form of\n$h(r)$ from \\eqref{eq:ch44Gtt}, as the $k(r)$ dependence cancels,\nand we find that $h(r)$ is given by:\n\\begin{equation} \\label{eq:ch6starhr}\nh(r) = \\left(1 + \\frac{r^{2}}{R^{2}} -\n\\frac{m(r)}{r^{d-3}}\\right)^{-1}\n\\end{equation}\nwhere $m(r)$ is a mass function\\footnote{Note that the mass function\nused here is a rescaling of the actual mass; a constant factor from\nthe integral over the angular directions does not appear in our\ndefinition of $m(r)$ due to our definition of $h(r)$.} related to\nthe density via:\n\\begin{equation} \\label{eq:ch6starmr}\nm(r) = \\frac{2}{d - 2} \\int_{0}^{r} \\rho(\\acute{r}) \\acute{r}^{d-2}\n\\, \\mathrm{d} \\acute{r}\n\\end{equation}\n\nIn order to specify a form for $k(r)$, we recall the energy-momentum\nconservation equation, $\\nabla_{\\mu} T^{\\mu \\nu} = 0$, which for a\ngeneral perfect fluid without the radiation condition gives:\n\\begin{equation} \\label{eq:ch6energymtmcons}\nP'(r) + \\frac{k'(r)}{2 \\, k(r)} (P(r) + \\rho(r)) = 0\n\\end{equation}\nwhich can be re-arranged to give\n\\begin{equation} \\label{eq:ch6starfr}\nk(r) = \\left(\\frac{\\rho_{\\infty}}{\\rho(r)} \\right)^{2\/d}\n\\end{equation}\nin the radiation case, where we have introduced $\\rho_{\\infty}$,\nwhich is the leading coefficient of $\\rho(r)$ at large $r$, and is\ngiven by $\\rho_{\\infty} \\approx \\rho(r) r^{d}$ as $r \\rightarrow\n\\infty$. Substituting $h(r)$ from \\eqref{eq:ch6starhr} into\n\\eqref{eq:ch44Grr} and eliminating $k'(r)\/k(r)$ using\n\\eqref{eq:ch6energymtmcons} then gives an equation in terms of\n$m(r)$, $\\rho(r)$ and $\\rho'(r)$,.\n\\begin{equation} \\label{eq:ch6starode1}\n\\frac{(d - 3)}{r^{2}} \\left(1 - \\frac{1}{1 + \\frac{r^{2}}{R^{2}} -\n\\frac{m(r)}{r^{d-3}}}\\right) - \\frac{2 \\rho'(r)}{r \\, \\rho(r) \\, d}\n= \\frac{2 \\rho(r) + (d-1)^{2}(d-2)\/R^{2}}{(d -1)(d-2) \\left(1 +\n\\frac{r^{2}}{R^{2}} - \\frac{m(r)}{r^{d-3}}\\right)}\n\\end{equation}\nwhich couples with our equation for $m'(r)$:\n\\begin{equation} \\label{eq:ch6starode2}\nm'(r) = \\frac{2}{d-2} \\, \\rho(r) \\, r^{d-2}\n\\end{equation}\nto give a pair of ODEs. For specified dimension $d$, these allow the\ngeometry of the spacetime to be numerically generated when they are\ncombined with the relevant boundary conditions: $m(0) = 0$ and\n$\\rho(0) = \\rho_{0}$. The condition $\\rho(0) = \\rho_{0}$ specifies\nthe central density of the gas, and we have that for fixed $R$,\n$\\rho_{0}$ is the single free parameter of the system (pure AdS is\nrecovered when $\\rho_{0} = 0$).\n\n\\section{Total mass as a function of central density}\n\\label{sec:analysis}\n\nWhilst mathematically one can work with this perfect fluid setup in\nany number of dimensions (including non-integer ones), one would\nalso like to consider the appropriateness of doing so, given that we\nwish to use the geometry as the setup for a toy model of a star. In\nother words, is there any significant change in behaviour as the\ndimensionality of the model is altered. A particular quantity of\ninterest in analysing the stability of the model is the total mass\n$M$ of the star, and as we have just seen, the mass and density\nprofiles of our gas of radiation are determined by a single\nparameter: the central density of the gas, $\\rho_{0}$.\n\nTo avoid possible instabilities such as those considered in the\nasymptotically flat case (in four dimensions) in \\cite{sorkin}, one\nwould expect the total mass to increase monotonically with\n$\\rho_{0}$. One could also expect the total mass to be bounded from\nabove by some maximum value, analogous to the $4 d$ asymptotically\nflat case where for a fixed size $R_{star}$, the maximum possible\nmass such a star can have is given by $M_{max} = 4 R_{star}\/9$, a\nresult found by Buchdahl in 1959 \\cite{buch}.\\footnote{For further\nedification, note that this equality coincides with the\nBuchdahl-Bondi limit, usually written in the form $R\/M = 9\/4$, which\nis the lowest radius to which the Schwarzschild geometry can be\nembedded (in Euclidean space). For more detail see e.g.\n\\cite{heinznew,abramnew}.}\n\n\\begin{figure}\n\\begin{center}\n \n  \\includegraphics[width=0.9\\textwidth]{figures\/alldmass2.eps}\\\\\n\\end{center}\n\\vspace{-1.0cm} \\caption{Total mass vs density for the radiating\nperfect fluid model in various dimensions, from $d=4$ (top curve)\nthrough to $d = 12$ (bottom curve). The saturation point for each\ndimension is indicated by the red dots; these correspond to the\nmaximum value of the total mass in the relevant dimension, at the\ncritical density $\\rho_{c}$ (see section \\ref{sec:criticald}). For\n$d$ large, there is no local maximum and hence no finite saturation\npoint; in these cases, the maximum total mass is given by the\nasymptotic value, $\\eta_{d}$. }\\label{sat1}\n\\end{figure}\n\nIn our scenarios the total mass is indeed bounded from above,\nhowever, this maximum is not always the asymptotic value of the\ntotal mass at large density. Although we observe that as $\\rho_{0}\n\\rightarrow \\infty$ we have $M(\\rho_{0}) \\rightarrow \\eta_{d}$,\nwhere $\\eta_{d}$ is some finite constant dependent on the dimension\n$d$ (see section \\ref{sec:masslargerho} below for more details),\nwhat we do not find in all cases is the total mass approaching this\nconstant monotonically, see figure \\ref{sat1}. When the\ndimensionality is low, there are sizable oscillations about the\nfinal value $\\eta_{d}$ before the curve settles down (see figure\n\\ref{lowdplot}), as was noted in the $d=5$ case in \\cite{hubenynew},\nand in other similar scenarios, e.g. \\cite{donp}, and the star's\nmaximum mass is given by some value greater than $\\eta_{d}$. As the\ndimension is increased, however, these oscillations become less\npronounced, and for $d$ sufficiently high they disappear altogether,\nsee figure \\ref{highdplot}.\n\n\\begin{figure}\n\\begin{center}\n \n  \\includegraphics[width=0.9\\textwidth]{figures\/lowdmass1c.eps}\\\\\n\\end{center}\n\\vspace{-1.0cm} \\caption{The oscillations in the total mass $M$: as\nthe central density $\\rho_{0}$ is increased, $M$ does not simply\nincrease monotonically towards its final value $\\eta_{d}$. Instead,\nit reaches a larger maximum before undergoing damped oscillations\ntowards $\\eta_{d}$. Note that the amplitude of the oscillations\nbecomes smaller as the dimensionality $d$ is\nincreased.}\\label{lowdplot}\n\\end{figure}\n\n\\begin{figure}\n\\begin{center}\n \n  \\includegraphics[width=0.9\\textwidth]{figures\/highdmass1c.eps}\\\\\n\\end{center}\n\\vspace{-1.0cm} \\caption{For larger $d$, there are no oscillations\nin the total mass: $M$ is now a monotonic function of the central\ndensity $\\rho_{0}$, and its maximum is also its asymptotic value as\n$\\rho_{0} \\rightarrow \\infty$, namely $\\eta_{d}$.}\\label{highdplot}\n\\end{figure}\n\nIdeally one would like to analytically determine the dependence of\nthe shape of the curve on both the dimension $d$ and the density\n$\\rho_{0}$, however, due to the complexity of the equations, the\nexact behaviour must be computed numerically. One can nevertheless\nuse this data to construct models of the various features of the\nstar's behaviour: for example, in section \\ref{sec:masslargerho}\nbelow, we give an analysis of the total mass at large $\\rho_{0}$\n(where it approaches a constant, dependent on $d$) in different\ndimensions.\n\nOne particularly interesting feature is the appearance of the\nturning points in the total mass seen in figure \\ref{lowdplot}, and\nspecifically the locations of the local maxima in different\ndimensions. One can see from the figures that as the dimensionality\nis increased, the appearance of the first maximum moves to larger\n$\\rho_{0}$; by analysing this progression one can obtain a\nremarkably simple relation which immediately gives a value for the\ncritical dimension, above which the oscillations do not exist, and\nhence the total mass is a monotonic function of $\\rho_{0}$.\n\n\\subsection{A critical dimension} \\label{sec:criticald}\n\nThe saturation point, $\\rho_{c}$, which we define as being the\nlocation of the first local maximum when increasing $\\rho_{0}$, can\nbe seen to progress to larger and larger $\\rho_{0}$ as the dimension\n$d$ is increased, see figure \\ref{sat1}. What we wish to determine\nis whether this saturation point appears for all dimension $d$ (for\nsufficiently large $\\rho_{0}$), or whether there is a cut-off\ndimension, $d_{c}$, such that for larger $d$, there is no local\nmaximum and hence no saturation point. Figure \\ref{peak1} shows how\nthe saturation point varies with dimension; numerical analysis then\nfinds (to 3 significant figures) that this behaviour is given by the\nfollowing model:\n\n\\begin{equation} \\label{eq:ch6peakest1}\n\\log{\\rho_{c}} \\approx 0.500 \\, d + \\frac{5.75}{\\sqrt{11.0 - d}} -\n2.20\n\\end{equation}\nwhich gives a critical dimension $d_{c} = 11.0$.\\footnote{As\nmentioned earlier, correspondence with V.~Vaganov and P.~H.~Chavanis\nsuggested that the critical dimension in the radiating perfect fluid\ncase is very close to (but not exactly) eleven, and this is indeed\nthe case as we see in the dynamical systems analysis approach in\nsection \\ref{sec:dynamsys}, where we obtain a value complementary to\nthe numerical estimate of $11.0$ given here. Interestingly, the\nexact value of $d_{c} = 11$ appears in the case of Newtonian\nisothermal spheres, as noticed by Sire and Chavanis in 2002\n\\cite{chan2}.} What is perhaps rather surprising is the simplicity\nof \\eqref{eq:ch6peakest1}: not only do we have a critical dimension\nappearing so clearly, the overall dependence on $d$ is remarkably\nsimple, and the co-efficient of the linear term appears to be\nexactly one half.\n\n\\begin{figure}\n\\begin{center}\n \n  \\includegraphics[width=0.9\\textwidth]{figures\/peakdatanew2.eps}\\\\\n\\end{center}\n\\caption{The change in the saturation point $\\rho_{c}$ with\nincreasing dimension $d$. The points plotted are the calculated\nvalues for the saturation point for the star model in the\ncorresponding dimension, the red best fit line is the curve given by\n\\eqref{eq:ch6peakest1}. The divergent behaviour as $d$ approaches\neleven indicates that for $d > 11$ there is no saturation point, and\nhence no apparent instability in the perfect fluid model of the\n``star''. }\\label{peak1}\n\\end{figure}\n\nAs we shall see in section \\ref{sec:dynamsys}, the value of the\ncritical dimension can also be determined by an analytical\nconsideration of the radiating perfect fluid system with zero\ncosmological constant (i.e. in the limit $R \\rightarrow \\infty$).\nAlthough such a solution is singular at $r = 0$, and has infinite\nmass, by confining the radiation to finite sized box one can obtain\nfinite mass solutions. The features determined in this configuration\ncan be related to equivalent behaviour in the asymptotically anti-de\nSitter case (where the (finite) mass is confined by the AdS\npotential), and indeed exact values for certain parameters can also\nbe computed. This same analysis is not restricted to the star\ngeometries considered here, it can be used with any linear equation\nof state \\cite{vaga}, or even more generally \\cite{uggla}. Before\ngiving the analysis for our case of perfect fluid radiation,\nhowever, we firstly present further numerical results.\n\n\\subsection{Total mass at large $\\rho_{0}$} \\label{sec:masslargerho}\n\nIn addition to considering the variation of the saturation point for\nthe star with dimension, one can also investigate the asymptotic\nbehaviour of $M$ as $\\rho_{0}$ becomes large. As mentioned in\nsection \\ref{sec:analysis}, at large $\\rho_{0}$, the value of the\ntotal mass tends to a constant, $\\eta_{d}$, which is then only\ndependent on the dimension; the value of this constant decreases as\n$d$ increases. The values are plotted in figure \\ref{etaplot1} and\ndespite the complicated appearance of the plot, a remarkably close\nfit for all dimensions is given by:\n\n\\begin{figure}\n\\begin{center}\n \n  \\includegraphics[width=0.45\\textwidth]{figures\/eta1c.eps}\n  \\includegraphics[width=0.45\\textwidth]{figures\/maxmass1bNEW.eps}\\\\\n\\end{center}\n\\caption{The plot on the left shows $\\eta_{d}$ for various\ndimensions, with the approximation given in \\eqref{eq:eta1} shown in\nred. The data points are all at integer values for the dimension,\nwith the addition of points at $d = 3.1, 3.2,...,3.5$ to highlight\nthe behaviour of the curve at low $d$. The righthand plot shows the\nbehaviour of $M_{d}^{max}$; this is identical to that of $\\eta_{d}$\nfor $d \\ge 11$, however for $d < 11$, the maximum is given by the\nvalue of the total mass at the saturation point, $\\rho_{c}$. The\nbest fit approximation (red curve) for each is simple in terms of\nits $d$ dependence, and provides a good fit over a large range of\n$d$.}\\label{etaplot1}\n\\end{figure}\n\n\\begin{equation} \\label{eq:eta1}\n\\eta_{d} \\approx 0.716 + \\exp \\left[9.85 - 3.72 \\, d \\right] - \\exp\n\\left[- 0.603 - \\frac{20.3}{d} \\right]\n\\end{equation}\nwhich is also shown in the figure. Checks show that the function\ncontinues to give accurate predictions for larger $d$, and although\nthere is perhaps slightly more complicated behaviour for $d \\sim 4$,\nwe do not attempt to investigate this further here\\footnote{Whilst\nin the dynamical systems analysis (section \\ref{sec:dynamsys}) the\n$d = 3$ case (where we have $\\eta_{3} = 1$) needs considering\nseparately, as there is different asymptotic behaviour involved due\nto the non-dynamical nature of gravity in such a scenario (see\n\\cite{hammer2} for example), we find that we can include it both\nhere (in the analysis of $\\eta_{d}$) and also in our earlier result\nfor the critical dimension (see section \\ref{sec:criticald}).};\ndespite the relative compactness of the expression, there is little\nintuitive origin for any of the constants involved. Nonetheless, it\nis impressive that the behaviour of the mass at large $\\rho_{0}$ can\nbe expressed in such simple powers of the dimension.\n\nOne can perform a similar analysis of the behaviour of the maximum\nvalue of the total mass as the dimension increases; the results are\nalso shown in figure \\ref{etaplot1}. For $d > 11$, the maximum total\nmass corresponds to the asymptotic value, $\\eta_{d}$, however, for\nlower dimension, the maximum is given by the mass at the saturation\npoint. A good fit to the curve is given by:\n\\begin{equation} \\label{eq:eta2}\nM_{d}^{max} \\approx 0.712 + \\exp \\left[2.74 - 1.07 \\, d \\right] -\n\\exp \\left[- 0.592 - \\frac{20.5}{d} \\right]\n\\end{equation}\nwhich differs (significantly) from \\eqref{eq:eta1} only in the\nsecond term; this was to be expected, as the curves differ only at\nlow $d$. Whilst their is little apparent significance about the\nvalues of the numerical constants involved in the expression, we\nhave again produced a fit with a relatively simple dependence on $d$\nwhich gives accurate predictions for $M_{d}^{max}$ over a large\nrange of dimensions.\n\nAlthough the form of the fit used in equations \\eqref{eq:eta1} and\n\\eqref{eq:eta2} was chosen primarily because it gave such a close\nfit to the data, it would be interesting to study possible reasons\nfor expecting the observed dependence on dimension, although we do\nnot pursue this further here. What we do now examine in more detail,\nis the oscillatory behaviour, which can be considered both\nnumerically and analytically.\n\n\\subsection{Self-similarity analysis for $d < 11$}\n\\label{sec:selfsim}\n\nAnother interesting feature of the plots of the total mass seen in\nfigures \\ref{sat1} and \\ref{lowdplot} is the self-similarity\nexhibited by the oscillatory behaviour as $\\rho_{0} \\rightarrow\n\\infty$. A numerical analysis of the periodicity and damping of the\noscillations seen for $3 < d < 11$ leads us to propose the following\nmodel for the total mass:\n\\begin{equation} \\label{eq:osc1}\nM_{d} (\\rho_{0}) \\approx \\eta_{d} + \\alpha_{d} \\exp \\left[-\n\\beta_{d} \\, \\log(\\rho_{0}) \\right] \\cos\\left[\\mu_{d} - \\nu_{d}\n\\log(\\rho_{0})\\right]\n\\end{equation}\nwhich gives a good approximation for the behaviour in the region\n$\\rho_{0} > \\rho_{c}$. In \\eqref{eq:osc1}, $\\eta_{d}$ is the\nasymptotic value of the mass discussed above, and the four\nparameters $\\alpha_{d}$, $\\beta_{d}$, $\\mu_{d}$ and $\\nu_{d}$ are\nconstants dependent only on the dimension $d$. Approximate values\nfor these constants for $d = 3.1, 4, 5, 6$ and $7$ are given in\nTable \\ref{addDtab1}.\n\n\\begin{table}\n\\begin{center}\n\\begin{tabular}{l l l l l}\n\\hline $d$ & $\\alpha_{d}$ & $\\beta_{d}$ & $\\mu_{d}$ & $\\nu_{d}$\\\\\n\\hline 3.1 & 0.305 & 0.184 & 8.33 & 0.66 \\\\\n4 & 0.383 & 0.371 & 8.44 & 0.86 \\\\\n5 & 0.400 & 0.601 & 9.35 & 0.98 \\\\\n6 & 0.415 & 0.825 & 10.3 & 1.03 \\\\\n7 & 0.431 & 1.03 & 11.4 & 1.07 \\\\\n\\hline\n\\end{tabular}\n\\end{center}\n\\caption{Numerical estimates for $\\alpha_{d}$, $\\beta_{d}$,\n$\\mu_{d}$ and $\\nu_{d}$ (to three significant figures) for the model\nof the total mass given in \\eqref{eq:osc1} .}\\label{addDtab1}\n\\end{table}\n\nAlthough the values given in table \\ref{addDtab1} are only\napproximate, we nonetheless see interesting dependencies on $d$\nemerging. For example, $\\beta_{d}$ appears to increase roughly\nlinearly with dimension ($\\beta_{d} \\approx 0.22 d - 0.51$), as do\n$\\alpha_{d}$ and $\\mu_{d}$ for $d \\ge 4$. We will see below in the\ndynamical systems analysis how this linear behaviour of $\\beta_{d}$\non $d$ is only an approximation to the true behaviour, and the same\nanalysis also provides exact values for the parameter $\\nu_{d}$.\nThis analytical analysis also confirms the form of the fit used in\nthe numerical approximation derived above.\n\n\\section{Dynamical systems analysis} \\label{sec:dynamsys}\n\nBy considering the behaviour of the system of coupled ODEs given in\nsection \\ref{sec:fluidmodels} in the limit $R \\rightarrow \\infty$,\nwe can obtain analytical results for some of the interesting\nfeatures of the radiating perfect fluid star geometries described\nabove. The analysis presented here follows that detailed in both\n\\cite{vaga} and \\cite{uggla}, where it is given in more general\nsettings; by focusing on the radiation case (where $\\rho(r) = (d-1)\nP(r)$) we can give a good explanation of why the numerical behaviour\nseen above is so, without excessive over-complication.\n\nThe basic idea is to rewrite the equations for $\\rho'(r)$ and\n$m'(r)$ in terms of dimensionless (compact) variables and perform an\nanalysis of the fixed points. The corresponding eigenvalues and\neigenvectors obtained by linearising about these fixed points give a\ncomplete description of the nearby behaviour (Hartman-Grobman\ntheorem, \\cite{hart}) on the new state space, which can then be\ntranslated back to the physical picture by inverting the\ntransformations given below. Interestingly, for the perfect fluid\nstars, the dependence of the total mass (as well as other\nquantities, e.g. the entropy) on the central density, $\\rho_{0}$, is\ngoverned by the behaviour around (and hence the eigenvalues of) a\nsingle fixed point. Specifically for our case we will see how this\ngives both an exact value for the critical dimension $d_{c}$, and a\nclear analytical explanation for the observed behaviour in the two\nregimes $d < d_{c}$ (oscillatory) and $d > d_{c}$ (monotonically\nincreasing). We will also obtain expressions for the $\\beta_{d}$ and\n$\\nu_{d}$ parameters introduced earlier.\n\nTo proceed, we thus set $R = \\infty$, and our equations\n\\eqref{eq:ch6starode1} and \\eqref{eq:ch6starode2} become:\n\\begin{equation} \\label{eq:ch6dynam1}\n\\rho'(r) = - \\frac{\\rho(r) \\, d \\left( (d-3)(d-2)(d-1) m(r) + 2\nr^{d-1} \\rho(r) \\right)}{2 (d-2)(d-1)(r^{d-2} - r \\, m(r))}\n\\end{equation}\n\\begin{equation} \\label{eq:ch6dynam2}\nm'(r) = \\frac{2}{d-2} \\, \\rho(r) \\, r^{d-2}\n\\end{equation}\nwhere $8 \\pi G_{d}$ has again been set equal to one. Note that we do\nnot include the $d = 3$ scenario here as it is a special case (due\nto the non-dynamical nature of gravity). We can now introduce the\ndimensionless variables:\n\\begin{equation} \\label{eq:ch6dynam3}\nu = \\frac{2 \\, r^{d-1} \\rho(r)}{(d-2) m(r)}\n\\end{equation}\nand\n\\begin{equation} \\label{eq:ch6dynam4}\nv = \\frac{(d-1) \\, m(r)}{2 \\, r^{d-3}} \\left(1 -\n\\frac{m(r)}{r^{d-3}} \\right)^{-1}\n\\end{equation}\nwhich allow equations \\eqref{eq:ch6dynam1} and \\eqref{eq:ch6dynam2}\nto be rewritten in the form:\n\\begin{equation} \\label{eq:ch6dynam5}\n\\frac{d u}{d \\xi} = - u \\left(1 - d + u + \\left(d - 3 +\n\\frac{u}{d-1}\\right)\\left(\\frac{v \\, d}{d-1}\\right)\\right)\n\\end{equation}\n\\begin{equation} \\label{eq:ch6dynam6}\n\\frac{d v}{d \\xi} = - v \\left(d - 3 - u + \\left(d - 3 - u \\right)\n\\left(\\frac{2 \\, v}{d-1}\\right)\\right)\n\\end{equation}\nwhere we have also introduced the new independent variable $\\xi =\n\\ln(r)$. For the case of positive mass and density we're considering\nhere, both $u$ and $v$ are greater than zero (for non-zero $r$), and\nwe make a final change of variables to the bounded $U$ and $V$\ndefined by\\footnote{Although the range of both $U$ and $V$ is\ndefined as being $(0,1)$, in order to perform the fixed point\nanalysis of the asymptotic behaviour, it is necessary that the\nboundary points also be included; this requires the system given by\n\\eqref{eq:ch6dynam8} and \\eqref{eq:ch6dynam9} be $\\mathcal{C}^{1}$\non $[0,1]^{2}$, which is manifestly so.}:\n\\begin{equation} \\label{eq:ch6dynam7}\nU = \\frac{u}{1+u} \\, , \\;\\;\\;\\;\\;\\; V = \\frac{v}{1+v}\n\\end{equation}\nwhich gives the system of equations:\n\\begin{equation} \\label{eq:ch6dynam8}\n\\frac{d U}{d \\lambda} = U (1-U) \\left( d-1- d U - \\left(2 d -4 +\n\\frac{d-3}{d-1} \\right) V + \\left(2 d - 3 + \\frac{d(d-5)\n+3}{(d-1)^{2}}\\right) U V \\right)\n\\end{equation}\n\\begin{equation} \\label{eq:ch6dynam9}\n\\frac{d V}{d \\lambda} = V (1-V)(3 - d + (d-2)\nU)\\left(1+\\left(\\frac{2}{d-1} - 1\\right) V\\right)\n\\end{equation}\nwhere we have also introduced the independent variable $\\lambda$,\ndefined by:\n\\begin{equation} \\label{eq:ch6dynam10}\n\\frac{d \\lambda}{d \\xi} = \\frac{1}{(1-U)(1-V)} \\;\\;\\;\\;\\; \\Bigg( =\n(1+u)(1+v) \\Bigg)\n\\end{equation}\n\nThe fixed points of the system (\\eqref{eq:ch6dynam8} and\n\\eqref{eq:ch6dynam9}) are calculated in the usual fashion, by\nsetting both $d U\/d \\lambda$ and $d V\/d \\lambda$ to zero and solving\nfor $U$ and $V$; there are six in total, with eigenvalues then\nobtained from\n\\begin{equation} \\label{eq:ch6dynam11}\n\\frac{d}{d \\lambda}\\left( \\begin{array}{c} U \\\\\nV \\end{array} \\right) = \\left( \\begin{array}{cc}\n\\frac{\\partial}{\\partial U} \\left(\\frac{d U}{d \\lambda}\\right) & \\frac{\\partial}{\\partial V} \\left(\\frac{d U}{d \\lambda}\\right)\\\\\n\\frac{\\partial}{\\partial U} \\left(\\frac{d V}{d \\lambda}\\right) &\n\\frac{\\partial}{\\partial V} \\left(\\frac{d V}{d \\lambda}\\right)\n\\end{array}\n\\right)\\Bigg|_{fp} \\left( \\begin{array}{c} U - U_{fp}\\\\\nV - V_{fp}\\end{array} \\right)\n\\end{equation}\nwhere the matrix components are evaluated at the particular fixed\npoint under consideration. A table of such eigenvalues is given in\n\\cite{vaga}, where they are labelled $T_{1}, \\dots, T_{6}$; we do\nnot list them all again here, however, as orbits in the interior of\nthe state space $[0,1]^{2}$ originate from either $T_{2}$ or $T_{4}$\nand converge to the fixed point $T_{3}$ (as is shown in\n\\cite{uggla}). This fixed point corresponds to the singular\nself-similar solution given by equation (2.14) of \\cite{vaga}, and\ndue to the scale invariance of the system one can consider the\nentire set of (positive mass) solutions as being represented by a\nsingle orbit from $T_{2}$ to $T_{3}$ (this is true for any linear\nequation of state, $P(r) = q \\rho(r)$).\n\nAlthough one cannot write an analytic expression for this orbit, one\ncan obtain approximations by linearising about the fixed points. As\ndiscussed briefly earlier, as this zero-cosmological constant\nsolution is singular, in order to produce finite mass solutions the\nradiation must be confined to an (unphysical) box; the two fixed\npoints $T_{2}$ and $T_{3}$ thus represent solutions with $\\rho_{0} =\n0$ and in the limit $\\rho_{0} \\rightarrow \\infty$ respectively. The\nbehaviour described by the linearisation about $T_{3}$ then reveals\naspects of the large $\\rho_{0}$ limit of the radiating stars (where\nthe confining AdS potential results in the finite mass solutions\nwithout the need for any unphysical box), exactly what we analysed\nnumerically in sections \\ref{sec:masslargerho} and\n\\ref{sec:selfsim}. This linearisation gives an explanation for the\nexistence of a critical dimension and the differing behaviour seen\nin higher and lower dimensions, including quantitative expressions\nfor $d_{c}$ and the $\\beta_{d}$ and $\\nu_{d}$ parameters of Table\n\\ref{addDtab1}, as we shall now show.\n\nFixed point $T_{3}$ corresponds to the following values of $U$ and\n$V$:\n\\begin{equation} \\label{eq:ch6dynam12}\nU_{T_{3}} = \\frac{d-3}{d-2} \\, , \\;\\;\\;\\;\\;\\; V_{T_{3}} = \\frac{2\n(d-1)^{2}}{2 - 4 \\, d + (d-1)d^{2}}\n\\end{equation}\nand has eigenvalues:\n\\begin{eqnarray} \\label{eq:ch6dynam13}\nT_{3}^{\\pm} && \\hspace{-0.5cm} = \\frac{d (d-3)}{2- 4 \\, d +\n(d-1)d^{2}}\\left( 1-d \\pm \\sqrt{\\frac{(d-12)d^{2} + 13 \\, d -\n18}{d-2}} \\right)  \\nonumber \\\\\n&& \\hspace{-0.5cm} \\equiv k_{d} \\left( 1-d \\pm\n\\sqrt{\\frac{(d-12)d^{2} + 13 \\, d - 18}{d-2}} \\right)\n\\end{eqnarray}\nwhere we denote the coefficient $k_{d}$ and observe that it is\nstrictly positive for $d > 3$. These eigenvalues govern the\nbehaviour of the solution, and we immediately see that there are two\ndistinct regimes; one where the expression inside the square root is\nnegative, corresponding to the oscillatory behaviour seen in figure\n\\ref{lowdplot}, and one where the expression is positive, resulting\nin the monotonic behaviour seen in figure\n\\ref{highdplot}.\\footnote{The fact that $k_{d} > 0$ ensures that the\nfixed point $T_{3}$ is a stable focus for the oscillatory behaviour\nin the $d < d_{c}$ case; for $d > d_{c}$, we have that $T_{3}^{\\pm}$\nis strictly less than zero, and hence acts as a stable node.} We\nthus obtain a value for the critical dimension given by the solution\nto:\n\\begin{equation} \\label{eq:ch6dynam14}\n(d_{c}-12)d_{c}^{2} + 13 \\, d_{c} - 18 = 0\n\\end{equation}\nwhich yields $d_{c} = 10.964\\dots$, complementary to the value of\n$d_{c} = 11.0$ obtained numerically, although with the significance\nof being non-integer rather than exactly $11$; interestingly for any\nlinear equation of state the value of $d_{c}$ is always in the range\n$10 \\le d_{c} \\le 11$, see \\cite{vaga}.\n\nWe can relate the asymptotic behaviour obtained from the state space\npicture to the physical quantities of mass and density via several\nauxiliary equations to those given above, specifically:\n\\begin{equation} \\label{eq:ch6dynam15}\n\\frac{d r}{d \\lambda} = (1 - U)(1 - V) r \\, , \\;\\;\\; \\frac{d m}{d\n\\lambda} = U(1 - V) m\n\\end{equation}\nand\n\\begin{equation} \\label{eq:ch6dynam16}\n\\frac{d \\rho}{d \\lambda} = - \\frac{V \\, d}{d-1}\\left(1 - U +\n\\frac{U}{d-1} \\right) \\rho\n\\end{equation}\n\nGiven expressions for $U$ and $V$ in terms of $\\lambda$ (as obtained\nfrom an analysis of the behaviour around the fixed points, say), one\ncan integrate the above to determine corresponding expressions for\nthe mass, radius and density in terms of $\\lambda$. There is,\nhowever, a simple way to see the dependence of the total mass\n$M_{d}$ on the central density $\\rho_{0}$, which also reveals the\norigin of the $\\beta_{d}$ and $\\nu_{d}$ parameters of our numerical\nmodel in section \\ref{sec:selfsim}.\n\nFocusing then on the case where $d < d_{c}$, how does the imaginary\nterm in \\eqref{eq:ch6dynam13} lead to the (self-similar) oscillatory\nbehaviour manifest in the total mass at large $\\rho_{0}$? This can\nbe seen directly from the linearisation about $T_{3}$, where we\nobserve similar oscillations in the expressions for $U(\\lambda)$ and\n$V(\\lambda)$ (see below); as mentioned above, it is the behaviour\naround $T_{3}$ that governs the behaviour of the physical quantities\nin the large $\\rho_{0}$ limit. By considering the behaviour of\n$U(\\lambda)$ and $V(\\lambda)$ in terms of $\\rho_{0}$, we can extract\nthe coefficients which should then match those in \\eqref{eq:osc1}\n(as argued more fully in \\cite{uggla}). The solutions of\n\\eqref{eq:ch6dynam8} and \\eqref{eq:ch6dynam9} in the large $\\lambda$\nlimit (i.e. about the fixed point $T_{3}$) can be expressed as:\n\\begin{equation} \\label{eq:ch6dynam15b}\nRe(U(\\lambda)) = U_{T_{3}} + \\exp\\left(- (d-1) k_{d} \\lambda\\right)\n\\cos\\left(k_{d} \\lambda \\sqrt{\\frac{18-(d-12)d^{2} - 13 \\, d}{d-2}}\n\\right)\n\\end{equation}\n\\begin{equation} \\label{eq:ch6dynam16b}\nRe(V(\\lambda)) = V_{T_{3}} + \\exp\\left(- (d-1) k_{d} \\lambda\\right)\n\\cos\\left(k_{d} \\lambda \\sqrt{\\frac{18 -(d-12)d^{2} - 13 \\, d}{d-2}}\n\\right)\n\\end{equation}\nwhere we have only kept the real angular term as we are only\ninterested in the period of the oscillations ($\\nu_{d}$) and the\ncoefficient of the damping ($\\beta_{d}$); the extra factors (namely\n$\\alpha_{d}$ and $\\mu_{d}$) cannot be extracted directly from this\nanalysis.\\footnote{Technically, for a solution of this form one\nshould first make a linear change of coordinates such that the\nmatrix on the RHS of \\eqref{eq:ch6dynam11} is in diagonal form. This\nonly manifests itself, however, as extra multiplicative constants\nwhich do not affect the decay term $\\beta_{d}$ or oscillation period\n$\\nu_{d}$, and hence can be ignored.}\n\nFor sufficiently high density stars (i.e. with large $\\rho_{0}$), we\nhave $\\lambda \\propto \\frac{1}{2} \\frac{d \\lambda}{d\n\\xi}\\big|_{T_{3}} \\log(\\rho_{0})$, and we thus obtain:\n\\begin{equation} \\label{eq:ch6dynam17}\n\\beta_{d} = \\frac{(d-1) k_{d}}{2 (1- U_{T_{3}})(1- V_{T_{3}})} =\n\\frac{d}{4} + \\frac{1}{2 \\, d} - \\frac{3}{4}\n\\end{equation}\nand\n\\begin{eqnarray} \\label{eq:ch6dynam18}\n\\nu_{d} && \\hspace{-0.5cm} = \\frac{k_{d}}{2 (1- U_{T_{3}})(1-\nV_{T_{3}})} \\sqrt{\\frac{18 -(d-12)d^{2} - 13 \\, d}{d-2}}  \\nonumber \\\\\n&& \\hspace{-0.5cm} = \\frac{1}{4 \\, d} \\sqrt{(d-2)(18 -(d-12)d^{2} -\n13 \\, d)}\n\\end{eqnarray}\nwhich give the values shown in Table \\ref{addDtab1exact}, provided\nas a comparison to the numerical estimates obtained in section\n\\ref{sec:selfsim}. We see that they match very closely, with any\ndiscrepancies most likely due to a combination of numerical\nimprecision in the original data for the mass at large $\\rho_{0}$\nand the use of oscillations at insufficiently large $\\rho_{0}$ for\nthe asymptotic dependence to be totally dominant.\n\n\\begin{table}\n\\begin{center}\n\\begin{tabular}{l l l}\n\\hline $d$ & $\\beta_{d}$ & $\\nu_{d}$\\\\\n\\hline 3.1 & $231\/1240 \\approx 0.186$ & $\\sqrt{695519}\/1240 \\approx 0.672$\\\\\n4 & $3\/8 = 0.375$ & $\\sqrt{47}\/8 \\approx 0.857$\\\\\n5 & $3\/5 = 0.6$ & $2 \\sqrt{6}\/5 \\approx 0.980$ \\\\\n6 & $5\/6 \\approx 0.833$ & $\\sqrt{13\/3}\/2 \\approx 1.04$\\\\\n7 & $15\/14 \\approx 1.07$  & $2 \\sqrt{215}\/5 \\approx 1.05$ \\\\\n\\hline\n\\end{tabular}\n\\end{center}\n\\caption{Exact values (alongside decimal equivalents) obtained from\nthe dynamical systems analysis for $\\beta_{d}$ and $\\nu_{d}$ for the\nmodel of the total mass \\eqref{eq:osc1}.}\\label{addDtab1exact}\n\\end{table}\n\n\\section{Discussion} \\label{sec:discussion}\n\nWhat we have seen in the above analysis is firstly the appearance of\na critical dimension ($d_{c} = 11.0$) from the simple requirement of\na stability condition on our perfect fluid model for a gas of\nradiation, and a numerical study of the behaviour in various\ndimensions. Whilst the appearance of oscillations in the variation\nof the total mass with the central density $\\rho_{0}$ in the lower\ndimensional cases had been noted before, we saw here that such\noscillations do not appear to persist in the higher dimensional\ncases (see figures \\ref{lowdplot} and \\ref{highdplot}). Not only do\nthe oscillations die down as the dimensionality is increased, by\nanalysing the progression of the saturation point, $\\rho_{c}$, we\nfind that they disappear completely for $d$ above a certain value.\nAlthough this value was calculated to be $11.0$ in the original\nnumerical analysis, the dynamical systems approach which followed in\nsection \\ref{sec:dynamsys} not only gave a more precise, non-integer\nvalue ($d_{c} = 10.964\\dots$, as first derived in \\cite{vaga}), but\nalso explained analytically why one sees a change from oscillatory\nto monotonic behaviour as one increases the dimension past this\npoint. The section concluded by demonstrating how the $\\beta_{d}$\nand $\\nu_{d}$ parameters of the numerical model could also be\nderived via this analytical approach.\n\nWhat is remarkable is that from a seemingly basic condition (that of\nmonotonicity in the variation of the total mass), one arrives at\nsuch a simple relation for the dependence of $\\rho_{c}$ on $d$,\nnamely equation \\eqref{eq:ch6peakest1}. Such simplicity could not\nhave been expected given the complex nature of the initial setup,\nwhich allows the spacetimes in question to be generated only\nnumerically from the coupled ODEs. The extensions\n\\cite{dema1,dema2,elsk} to the BKL work on modelling a gravitational\nfield close to a spacelike singularity also reveal a critical\ndimension of eleven.\\footnote{Briefly, their analysis of the setup\nwas performed using the mixmaster model, where the dynamical\nbehaviour is governed by Kasner exponents and conditions upon them,\nand in which the evolution continues until the system reaches a\nstability region where the Kasner exponents remain constant. They\nobserved that such a stability region could only exist for $d \\ge\n11$, and thus the evolution continues indefinitely for any lower\nnumber of dimensions. This has interesting consequences not\nelaborated on here, which are discussed in detail in the papers\ncited above.} In their work they found that the general behaviour of\nthe relevant Einstein solutions changed from ``chaotic'' in the low\ndimensional cases ($d < 11$) to non-chaotic in higher dimensions ($d\n\\ge 11$), in much the same manner as we observe the transition from\noscillatory to monotonic total mass behaviour in the radiating star\ncase considered here. It is interesting that their work also reveals\na critical dimension of eleven, and a more detailed comparison of\nthe two different scenarios (including an analysis to determine the\nexact (possibly non-integer) value of $d_{c}$ for their transition)\nmay yield further insight.\n\nIt would be interesting to see if such a result appears in other\ninvestigations into scenarios similar to the radiating perfect fluid\nmodel considered here. For example, one could examine other physical\nequations of state to see if they exhibit the same behaviour, and\nindeed the work of \\cite{vaga} and \\cite{chan} has pursued this idea\nfurther (see the note below). Finally, one could also look to\nexplain the linear $d\/2$ term which appears in\n\\eqref{eq:ch6peakest1}, and whether there is any physical\nexplanation for why the coefficient should take on the value of a\nhalf.\n\n\\subsection*{Note}\n\nUnknown to the author, this phenomenon has also been simultaneously\ninvestigated in two other works. In \\cite{vaga}, Vladislav Vaganov\nanalyses the behaviour of radiating perfect fluid models in\n$d$-dimensional AdS spacetimes; he notes (as we do here) that there\nis a significant change in the behaviour of the total mass for $d >\n11$ (where it becomes a monotonic rather than oscillatory function\nof the central density), and demonstrates this not only for the mass\nbut also the temperature and entropy.\n\nHe also presents a dynamical systems analysis (based on that given\nin \\cite{uggla}) of the behaviour for a general linear equation of\nstate, $P(r) = q \\rho(r)$, which includes the radiation case. This\nanalysis complements the numerical results presented here, providing\nan analytic derivation of the critical density, which is determined\nto be $d_{c} = 10.964\\dots$, consistent with our relation\n\\eqref{eq:ch6peakest1}. The specific analysis for the radiation case\nis given in section \\ref{sec:dynamsys}, where we give not only the\nderivation of the critical density, but also demonstrate how the\ndynamical systems technique gives analytical expressions for other\nparameters introduced in our numerical investigation into the\nself-similar behaviour for $d < d_{c}$.\n\nThe second related paper, \\cite{chan} by Pierre-Henri Chavanis,\npresents an in-depth study of the behaviour of general stars\n(``isothermal spheres'') with a linear equation of state in an\nasymptotically flat background. His results are again complementary,\nfinding that there is monotonic behaviour for $d \\ge 11$, in\ncontrast to the oscillatory behaviour observed in lower dimensions.\nBy asymptotic analysis he also finds the value for the critical\ndimension in the radiation case to be very close to eleven, and\nalthough there initially appeared to be a discrepancy between the\ntwo alternative calculations of the critical dimension in\n\\cite{vaga} and \\cite{chan}, the latter was subsequently corrected\nto agree with the value of $d_{c} = 10.964\\dots$ found in\n\\cite{vaga}. His paper also includes a comprehensive investigation\ninto the stability of the different regimes, looking at a number of\nalternative stellar configurations and considering the behaviour of\nother thermodynamic parameters (entropy, temperature,$\\dots$), in\naddition to the mass.\n\n\n\\section*{Acknowledgements}\n\nI'd like to thank Veronika Hubeny for useful discussions and ideas,\nand Don Page for his comments which helped motivate this work, which\nwas supported by an EPSRC studentship grant and the University of\nDurham Department of Mathematical Sciences.\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
{"text":"\n\\section{Introduction and Statement of Main Result}\n\n\\subsection{Motivation}\n\nThe rate of convergence of the Green's function for different kinds of planar, discrete-time random walks to the continuous Green's function has been studied in a number of papers in general classes of domains (see for instance \\cite{kl}, \\cite{bjk}, and \\cite{jiangkennedy}), as well as in some specific domains, in the case of the simple random walk Green's function, such as the disk (see \\cite{greenbook}) and the half- and quarter-plane (see \\cite{spitzer}). The currently available results suggest that in the case of smooth domains, the rate of convergence should be of the same order as that of the lattice spacing, whereas in arbitrary domains, it should be of order square root of the lattice spacing. The present paper suggests an answer to the question of how the rate of convergence depends on domain regularity by examining a family of domains with one singular boundary point.\n\nThe question of rate of convergence of discrete Green's functions is intimately related to the question of the rate of convergence of discrete harmonic measure (see \\cite{SLEbook} for a discussion of the relation between the Green's function, harmonic measure, and the Poisson kernel, the Radon-Nikodym derivative of harmonic measure with respect to Lebesgue measure). It is shown in \\cite{jiangkennedy} that the rate of convergence  in smooth domains of harmonic measure for the discrete-time, continuous-space random walk considered in that paper is the same as the rate of convergence of the corresponding Green's function, that is, roughly the inverse of the step size. However, no equivalent result is currently available for simple random walk. The work done in the present paper is intended to be a first step towards obtaining rates of convergence for simple random walk harmonic measure.\nNote that discrete harmonic measure is known to converge in the domains we are examining in this paper (see, e.g., \\cite{lawlerlimic}, \\cite{chelkaksmirnov}, and \\cite{biskuplouidor}, which all discuss classes of domains which contain the domains considered in the present paper).\n\nThe importance of harmonic measure itself is manifold, but particularly obvious in the context of the Dirichlet problem: The Dirichlet problem in a domain $D\\subset\\C$ consists of finding a function $f:\\bar{D}\\to\\R$, harmonic in $D$, with prescribed boundary values $h$, that is, of finding a function $f:\\bar{D}\\to\\R$ such that \n\\begin{equation}\\label{dirichlet}\n\\left\\{\n\\begin{array}{ll}\n  \\Delta f(z)  \n \n  =0, & z\\in D\\\\\n  f(z) = h(z), & z\\in\\partial D.\n\\end{array}\n\\right.\n\\end{equation}\nWhether the Dirichlet problem actually has a solution depends both on $D$ and $h$. It is known (see \\cite{garnett_marshall} for an analytic point of view and \\cite{SLEbook} for a probabilistic approach) that if $D$ is a regular domain (this roughly means that all points in $\\partial D$ are part of a piece of a curve $\\subset \\partial D$ containing more than one point; in particular all simply connected domains are regular) and $h$ is continuous and bounded, then there exists a unique bounded, continuous solution to the Dirichlet problem. In that case, one can write the solution of \\eqref{dirichlet} as\n$$f(z)=\\int_{\\partial D} h(w)\\omega(z,|dw|; D),$$\nwhere $\\omega=\\omega(z, \\cdot; D)$ is harmonic measure from $z$ on $\\partial D$.\n\n\n\n\\subsection{Definitions and Important Properties}\n\nFor a domain $D\\subsetneq \\Z^2$, if $S$ is simple random walk started at $z$ and\n\\begin{equation}\\label{T}\nT_D=\\min\\{k\\geq 0:S_k\\not\\in D\\},\n\\end{equation} \nis the first time $S$ leaves $D$, the discrete Green's function in $D$ is, for $z,w\\in\\Z^2$,\n$$G_D(z,w)=E^z\\left[\\sum_{k\\geq 0}\\mathbbm{1}\\{S_k=w; k <T_D\\}\\right],$$\nthe expected number of visits to $w$ before leaving $D$ by $S$ started at $z$. We will write $G_D(w)=G_D(0,w)$.\n\nA representation of $G_D$ which will be particularly useful for us is the following (see \\cite{greenbook}): For $z,w\\in\\Z^2$,\n\\begin{equation}\\label{discretegreenrep}\nG_D(z,w)=E^z[a(S(T_D)-w)]-a(z-w),\n\\end{equation}\nwhere for $x\\in\\Z^2$, \n$$a(x)=\\sum_{j\\geq 0}(P^0(S(j)=0)-P^x(S(j)=0))$$\nis the potential kernel for simple random walk. As $|x|\\to\\infty$, $a$ has the representation\n\\begin{equation}\\label{ax}\na(x)=\\frac{2}{\\pi}\\log|x|+k_0+\\bigo{|x|^{-2}},\n\\end{equation}\nwhere $k_0=\\frac{2\\gamma+3\\ln 2}{\\pi}$ and $\\gamma$ is Euler's constant. See \\cite{FukU1} for more details.\n\nOne can define a continuous analogue of the discrete Green's function.\nIf $D\\subsetneq \\C$ is a domain such that for any $z\\in D$, if $B$ is standard Brownian motion started at $z$, \n\\begin{equation}\\label{tau}\n\\tau_D=\\inf\\{t\\geq 0:B_t\\not\\in D\\}\n\\end{equation}\nsatisfies $\\tau_D<\\infty, \\text{a.s.}$, we can define\n$$p_D(t,z,w)=\\lim_{\\epsilon\\to 0}\\frac{1}{\\pi\\epsilon^2}P^z(|B_t-w|\\leq \\epsilon, t\\leq \\tau_D),$$\nthe transition density for $B$ from $z$ to $w$ before exiting $D$. The Green's function in $D$ is then, for $z\\neq w$, \n$$g_D(z,w)=\\pi\\int_0^{\\infty} p_D(t,z,w)\\,dt.$$\nIt is the unique harmonic function on $D\\setminus \\{z\\}$ satisfying $\\lim_{w\\to w_0}g_D(z,w)=0$ for every regular (see \\cite{SLEbook} for a precise definition) boundary point $w_0\\in \\bd D$ and\n$$g_D(z,w) + \\log|z-w| = \\bigo{1}, \\qquad \\text{ as } |z-w|\\to 0$$\n(see below for the definition of $\\bigo{\\cdot}$). We will write $g_D(w)=g_D(0,w)$.\n\nAn analogue of \\eqref{discretegreenrep} holds for $g_D$: For $z,w\\in D$,\n\\begin{equation}\\label{contgreenrep}\ng_D(z,w)=E^z[\\log |B(\\tau_D)-w|]-\\log |z-w|.\n\\end{equation}\nNote that \\eqref{contgreenrep} implies that $g_D$ is unchanged under re-parametrizations of $B$.\n\nThe fundamental property of conformal invariance of planar Brownian motion carries over to the Green's function: If $\\psi:D\\to D'$ is a conformal transformation, then \n$$g_D(z,w)=g_{\\psi(D)}(\\psi(z),\\psi(w)).$$\n\n\n\n\n\n\nThroughout this paper, when we write $f(z)=\\bigo{g(z)}$, we mean that there exists a constant $c$ such that for all $z$ in a set which will depend on the context (usually, it will be for $z$ large enough or for $z$ small enough), $|f(z)|\\leq c|g(z)|$. We will use this notation for real-valued functions only and make it explicit when we need it for complex-valued functions. We will also use the notation $f(z)\\lesssim g(z)$ to mean the same thing and $f(z)\\gtrsim g(z)$ to mean $g(z)=\\bigo{f(z)}$. $f(z)=o(g(z))$ will mean $\\lim_{|z|\\to\\infty}f(z)\/g(z)=0$.\n\n\n\nThe proof of Theorem A.1 in \\cite{bjk} suggests that the worst-case scenario in arbitrary domains arises when the boundary contains a slit and that the rate of convergence of the Green's function is fastest when the domain is smooth.\nIt is therefore natural to consider the domains (see Figure \\ref{Fig1}\n\\begin{equation}\\label{da}\nD_\\alpha = D_{\\alpha}(n)=\\{re^{i\\theta}\\in \\C:0 <\\theta<2\\pi-\\alpha, 0<r<2n\\}-z_0,\n\\end{equation}\nwhere $0\\leq \\alpha\\leq \\pi$ and $z_0\\in\\Z^2$ is a point closest to $ne^{i(\\pi-\\alpha\/2)}$. Note that these domains have inner radius within one unit of $n$.\n\n\\subsection{Rate of Convergence of Discrete Green's Function}\n\nThis paper's main result is an upper bound for the rate of convergence of $G_{D_{\\alpha}}(w)$ for $w\\in D_{\\alpha}$:\n\n\n\\begin{thm}\\label{green}\nIf $w\\in D_{\\alpha}$,\n$$\\left|G_{D_{\\alpha}}(w)-\\frac{2}{\\pi}g_{D_{\\alpha}}(w)\\right|=\\bigo{\\left(\\frac{\\log^2 n}{n}\\right)^{c_{\\alpha}}+|w|^{-2}},$$\nwhere \n\\begin{equation}\\label{ca}\nc_{\\alpha}=\\frac{1}{2}+\\frac{\\alpha}{4\\pi-2\\alpha}=\\frac{\\pi}{2\\pi-\\alpha}.\n\\end{equation}\nIn particular, for $|w|\\geq (n\/\\log^2 n)^{c_\\alpha\/2}$,\n$$\\left|G_{D_{\\alpha}}(w)-\\frac{2}{\\pi}g_{D_{\\alpha}}(w)\\right|=\\bigo{\\left(\\frac{\\log^2 n}{n}\\right)^{c_{\\alpha}}}.$$\n\\end{thm}\n\n\n\n\n\\section{Proof of Theorem \\ref{green}}\n\n\\subsection{Exit Probabilities for $S$ and $B$ in $D_\\alpha$}\n\nCentral to our proof is a coupling between simple random walk and Brownian motion, called the KMT approximation (see \\cite{kmt2} and \\cite{benesnotes} for a simple argument justifying the extension of the result from dimension one to two) of which we state a consequence in Lemma \\ref{kmt} below. In this coupling of planar simple random walk and standard planar Brownian motion, it is random walk at time $2k$ and Brownian motion at time $k$ which are close to each other with high probability. For notational convenience, for the rest of this paper, we let $\\tilde{B}$ be standard planar Brownian motion and define for all $t\\geq 0$,\n\\begin{equation}\\label{reparam}\nB(t) = \\tilde{B}(t\/2).\n\\end{equation}\nNote that changing the speed of Brownian motion linearly doesn't affect its path properties and, as mentioned before, leaves the corresponding Green's function unchanged.\nWe also consider $S$ to be interpolated linearly between integer times, that is, for all $t\\in \\R_+$, \n\\begin{equation}\\label{interpol}\nS_t=S_{\\lfloor t\\rfloor}+(t-\\lfloor t\\rfloor)(S_{\\lceil t\\rceil}-S_{\\lfloor t\\rfloor}).\n\\end{equation}\nWith these definitions, we have\n\\begin{lem}\\label{kmt}\nThere exist $c_0$ and a probability space containing a planar Brownian motion $B$ as in \\eqref{reparam} and a two-dimensional simple random walk $S$ as in \\eqref{interpol} such that for all $n$, if $D$ is any set with outer radius at most $3n$, that is, such that $\\sup\\{|z|:z\\in D\\}\\leq 3n$, then for any $z\\in D$,\n$$P^z\\left(\\sup_{0\\leq t\\leq T_D\\vee\\tau_D}|S_t-B_t| > c_0\\log n\\right)=\\bigo{n^{-10}},$$\nwhere $P^z$ is the measure associated with $B$ and $S$ both started at $z$, $T_D$ is as in \\eqref{T}, and $\\tau_D$ is as in \\eqref{tau}, but for the reparametrized Brownian motion.\n \n\\end{lem}\n\n\n\n\n\nThe following basic estimates will be helpful in obtaining our key estimates below. The first and third follow from estimates for standard Brownian motion that can be found, e.g., in \\cite{benesnotes} and the second is obvious.\n\n\\begin{lem}\\label{toofar} If $B$ is planar Brownian motion as in \\eqref{reparam} and $S$ is two-dimensional simple random walk, then there exist constants $C$ and $K$ such that \n\\begin{enumerate}\n\\item[(a)] $P(\\sup_{0\\leq t \\leq 1}|B(t)| \\geq r) \\leq Ce^{-r^2}.$\n\\item[(b)] $P(\\sup_{0\\leq t \\leq 1}|S(t)| \\geq r) = 0 \\text{ if } r>1.$\n\\item[(c)] $P(\\sup_{0\\leq t \\leq n}|B(t)| \\leq r^{-1}n^{1\/2}) \\leq \\exp\\{-K r^2\\}.$\n\\end{enumerate}\n\\end{lem}\n\n\nAt the center of our argument are the following lemmas which estimate the probability of $B$ or $S$ leaving $D_{\\alpha}$ in some small subset of the boundary. The first follows from conformal invariance of planar Brownian motion and the second is obtained from the first using the KMT coupling of Lemma \\ref{kmt}.\n\nRecall the definition of $D_\\alpha$ in \\eqref{da} and of $z_0$ in the following line. Let\n$$N=\\lceil 2n\/\\log^2 n \\rceil$$\nand define for $1\\leq k\\leq N$,\n$$\nI_k=\\{z\\in\\bd D_\\alpha: (k-1)\\log^2 n\\leq |z+z_0| < k\\log^2 n\\}.\n$$\nNote that since by $\\log n$ we mean the natural logarithm, $n\/\\log^2 n$ cannot be an integer, so that $I_N$ contains the circular part of  $\\bd D_{\\alpha}$. Recall the definition of $c_\\alpha$ in \\eqref{ca}.\n\n\n\n\n\n\\begin{lem}\\label{beurling1} \nFix $\\alpha\\in [0,\\pi]$ and $a>0$. Then for all $n$ large enough,\nall $x\\in D_{\\alpha}$ satisfying $d(x,\\bd D_{\\alpha})\\leq a\\log n$ and $d(x,\\bd D_{\\alpha})=d(x,I_{k_0})$, and all $1\\leq k\\leq N$ for which $|k_0-k|\\geq 2$, \n$$P^x(B_{\\tau_{D_{\\alpha}}}\\in I_k)\\lesssim \\frac{(k_0k)^{c_{\\alpha}-1}}{(k^{c_{\\alpha}}-k_0^{c_{\\alpha}})^2\\log^{c_\\alpha} n}.$$\n\\end{lem}\n\n \\begin{figure}[htb!]\n\\centering%\n\\includegraphics[scale=0.84]{Pacman.pdf}\n\\caption\n$D_{\\alpha}$, together with $I_k$ for some arbitrary $1<k<N$. The dashed line is the set of points in $D_{\\alpha}$ at distance $2c_0\\log n$ from $\\bd D_{\\alpha}\ncorresponding to the hitting time $\\eta$ in the proof of Theorem \\ref{green}.}\n\\label{Fig1}\n\\end{figure} \n\n\n\\begin{proof} We will assume without loss of generality that \n$x$\nsatisfies $\\arg (x+z_0)\\in [0,\\pi-\\alpha\/2]$, in other words, is in the ``top half\" of $\\bd D_{\\alpha}$. \n\n\nWe will consider first the case where $k<N, k_0<N$. Note that the map $f=f_\\alpha$ defined by \n$$f_{\\alpha}(z)=\\left(\\frac{z+z_0}{2n}\\right)^{c_{\\alpha}}$$\nsends the domain $D_{\\alpha}$ to the unit upper half-disk \n\\begin{equation}\\label{d+}\n\\D^+=\\{z\\in\\C:\\text{Im}(z)>0, |z|<1\\}\n\\end{equation} \nand satisfies \n$$f(0)=\\left(\\frac{1}{2}\\right)^{c_\\alpha}i\\left(1+\\bigo{n^{-1}}\\right).$$ \nIt is also easy to verify that if for $2\\leq k_0\\leq N-1$, $d(x,\\bdD_{\\alpha})=d(x,x_0)$ with $x_0\\in  I_{k_0}$, then\n$$f(x)=\\frac{x_0^{c_{\\alpha}}}{(2n)^{c_{\\alpha}}}\\left(1+(1+i)\\bigo{\\frac{\\log n}{x_0}}\\right),$$\n and if $d(x,\\bd D_{\\alpha})=d(x,I_1)$, then\n$$0\\leq \\text{Re}(f(x))\\leq \\left(\\frac{\\log^2 n}{n}\\right)^{c_{\\alpha}}, \\quad 0\\leq \\text{Im}(f(x)) \\lesssim \\left(\\frac{\\log n}{n}\\right)^{c_{\\alpha}}.$$\nMoreover,\n$$f(I_k)=\\left(-\\left(\\frac{k}{2n}\\right)^{c_{\\alpha}}\\log^{2c_{\\alpha}}n, -\\left(\\frac{k-1}{2n}\\right)^{c_{\\alpha}}\\log^{2c_{\\alpha}}n\\right] \\cup \\left[\\left(\\frac{k-1}{2n}\\right)^{c_{\\alpha}}\\log^{2c_{\\alpha}}n, \\left(\\frac{k}{2n}\\right)^{c_{\\alpha}}\\log^{2c_{\\alpha}}n\\right).$$\nThen, with $\\D+$ as in \\eqref{d+}, for all $x$ with $d(x,\\bdD_{\\alpha})\\leq a\\log ^2 n, d(x,\\bdD_{\\alpha})=d(x,I_{k_0})$ for some $1\\leq k_0<N$, \n\\begin{eqnarray}\\label{gen}\\notag\n P^x\\left(B(\\tau_{D_\\alpha})\\in I_k\\right) &=& P^{f(x)}\\left(B(\\tau_{\\D^+})\\in f(I_k)\\right) \\leq  P^{f(x)}\\left(B(\\tau_{\\mathbbm{H}})\\in f(I_k)\\right)\\\\ \\notag\n& \\lesssim & P^{i}\\left(B(\\tau_{\\mathbbm{H}})\\in \\left[\\log^{c_\\alpha} n k_0^{1-c_{\\alpha}}|(k-1)^{c_{\\alpha}}-k_0^{c_{\\alpha}}|,\\log^{c_\\alpha} n k_0^{1-c_{\\alpha}}|k^{c_{\\alpha}}-k_0^{c_{\\alpha}}|\\right)\\right)\\\\ \n& \\lesssim & \\frac{(k_0k)^{c_{\\alpha}-1}}{(k^{c_{\\alpha}}-k_0^{c_{\\alpha}})^2\\log^{c_\\alpha} n}.\n\\end{eqnarray}\nwhere the equality follows from conformal invariance of planar Brownian motion, the first inequality follows from translation and scaling invariance of Brownian motion and the last inequality follows from the fact that for Brownian motion started at $i$, $B(\\tau_{\\mathbbm{H}})$ has the Cauchy distribution, the equality $\\arctan(x)+\\arctan(1\/x)=\\frac{\\pi}{2}$, and the Taylor expansion of $\\arctan$ at the origin. In fact, the computation yields the better bound of $\\frac{(k_0k)^{c_{\\alpha}-1}}{(k^{c_{\\alpha}}-k_0^{c_{\\alpha}})^2\\log n}$ for all $k_0\\geq 2$, but not for $k_0=1$.\n\nConsider now the case where $k_0=N$ and let $t_{x_0}$ be the tangent line to $\\bd D_{\\alpha}$ at $x_0$, a closest point to $x$ in $\\bdD_{\\alpha}$ and $\\mathbbm{H}_{x_0}$ the half-plane with boundary $t_{x_0}$ containing the origin. Then \n\\begin{equation}\\label{d}\nd:=\\inf\\{|w-x_0|:w\\in I_k\\}\\gtrsim (N-k)\\log^2 n. \n\\end{equation}\nThe strong Markov property applied at time $\\tau_{D\\left(x_0,\\frac{d}{2}\\right)}$ implies that\n\\begin{equation}\\label{12-3}\nP^x(B(\\tau_{D_\\alpha})\\in I_k)  \\leq  P^x\\left(\\tau_{D\\left(x_0,\\frac{d}{2}\\right)\\cap D_{\\alpha}} < \\tau_{D_{\\alpha}}\\right)\\sup_{z\\in D\\left(x_0,\\frac{d}{2}\\right)}P^z(B(\\tau_{D_\\alpha})\\in I_k).\n\\end{equation}\nIf $x_k^{(1)}, x_k^{(2)}$ are the midpoints of the two segments forming $I_k$, we can use the fact that since $x_0\\in I_N, \\max\\{d,x_k^{(i)}\\}\\asymp n$ for $i=1,2,$ to easily verify that there is a constant $C$ such that for $i=1, 2$, \n$$f\\left(D\\left(x_k^{(i)},\\frac{d}{2}\\right)\\cap D_\\alpha\\right) \\supset D\\left(f(x_k^{(i)}),C\\frac{d}{n}\\right)\\cap\\D_+.$$\nTherefore, since the width of each segment of $f(I_k)$ is \n$$f(k\\log^2 n-z_0)-f((k-1)\\log^2 n-z_0) \\lesssim \\left(\\frac{k\\log^2 n}{2n}\\right)^{c_\\alpha}k^{-1},$$ we have\n\\begin{eqnarray}\\label{second} \\notag\n\\sup_{z\\in D\\left(x_0,\\frac{d}{2}\\right)}P^z(B(\\tau_{D_\\alpha})\\in I_k) & \\leq & \\sup_{z\\in D\\left(x_k^{(1)},\\frac{d}{2}\\right)^c\\cap D\\left(x_k^{(2)},\\frac{d}{2}\\right)^c\\cap D_{\\alpha}}P^z(B(\\tau_{D_\\alpha})\\in I_k)\\\\ \\notag\n&\\lesssim & \\sup_{z\\in D\\left(f(x_k^{(1)}),\\frac{d}{n}\\right)^c\\cap D\\left(f(x_k^{(2)}),\\frac{d}{n}\\right)^c\\cap\\D_+}P^z(B(\\tau_{\\mathbbm{H}})\\in f(I_k))\\\\\n&\\lesssim & \\frac{\\left(\\frac{k\\log^2 n}{n}\\right)^{c_\\alpha}k^{-1}}{dn^{-1}}\\lesssim \\left(\\frac{k}{N}\\right)^{c_\\alpha-1}\\frac{\\log^2 n}{d},\n\\end{eqnarray}\nwhere the penultimate \ninequality follows from the fact that the exit distribution of the half-plane has the Cauchy distribution.\n\n\nMoreover, \n\\begin{equation}\\label{first}\nP^x\\left(\\tau_{D\\left(x_0,\\frac{d}{2}\\right)\\cap D_{\\alpha}} < \\tau_{D_{\\alpha}}\\right)  \\leq  P^x\\left(\\tau_{D\\left(x_0,\\frac{d}{2}\\right)\\cap \\mathbbm{H}_{x_0}} < \\tau_{D_{\\alpha}}\\right) \\lesssim \\frac{\\log n}{d},\n\\end{equation}\nwhere the second inequality is essentially the Gambler's ruin estimate but can be shown rigorously, via conformal invariance, using the fact that the map $-(z+z^{-1})$ is a conformal transformation of the upper unit half-disk into the upper half-plane and, again, the fact that the exit distribution of the half-plane has the Cauchy distribution.\n\nPlugging \\eqref{second} and \\eqref{first} into \\eqref{12-3} and using \\eqref{d} now implies that\n\\begin{equation}\\label{extreme}\nP^x(B(\\tau_{D_\\alpha})\\in I_k) \n\\lesssim \\left(\\frac{k}{N}\\right)^{c_\\alpha-1}\\frac{1}{(N-k)^2\\log n}.\n\\end{equation}\n\nThe lemma now follows from the fact that the bound in \\eqref{extreme} is of order at most that in \\eqref{gen} when $k_0=N$.\n\n\n\\end{proof}\n\n\n\n\\begin{lem}\\label{beurling2} Fix $\\alpha\\in [0,\\pi]$ and $a>0$. Then for all $n$ large enough,\nall $x\\in D_{\\alpha}\\cap\\Z^2$ satisfying $d(x,\\bd D_{\\alpha})\\leq a\\log n$ and $d(x,\\bd D_{\\alpha})=d(x,I_{k_0})$, and all $1\\leq k\\leq N$ for which $|k_0-k|\\geq 2$, \n$$P^x(S_{T_{D_{\\alpha}}}\\in I_k)\\lesssim \\frac{(k_0k)^{c_{\\alpha}-1}}{(k^{c_{\\alpha}}-k_0^{c_{\\alpha}})^2\\log^{c_\\alpha} n}.$$\n\\end{lem}\n\n\n\\begin{proof}\nWe use the KMT coupling of Lemma \\ref{kmt} to derive this estimate from the analogous estimate for Brownian motion in Lemma \\ref{beurling1}.\n\nAssume first that $\\alpha\\neq 0$ and define \n$$z_0'=z_0-2c_0\\log n(\\cot(\\alpha\/2)-i).$$\nNote that $-z_0'\\not\\inD_{\\alpha}$ is the point of intersection of two lines that are parallel to the segments of $\\partial D_{\\alpha}$ and at distance $2c_0\\log n$ of those segments. We then define\n$$D_{\\alpha}' = \\{z\\in\\C:|z+z_0|\\leq 2n+2c_0\\log n\\}\\setminus \\{x\\in\\C:\\arg(z+z_0')\\in (-\\alpha,0)\\}$$\nand, for $1\\leq k\\leq N$, \n$$I'_k=\\{z\\in\\bd D'_{\\alpha}:d(z,\\bdD_{\\alpha})=d(z,I_k)\\}.$$ \nWe assume for the rest of the proof that $n$ is large enough so that $I'_k\\neq\\emptyset$ for all $1\\leq k\\leq N$\n\n\nWe now couple $B$ and $S$ as in Lemma \\ref{kmt} and define \n\\begin{equation}\\label{k}\n\\mathcal{K}=\\left\\{\\sup_{0\\leq t\\leq T_{D_{\\alpha}}\\vee\\tau_{D_{\\alpha}}}|S_{t}-B_t| \\leq  c_0\\log n\\right\\}.\n\\end{equation}\nThen, by Lemma \\ref{kmt}, \n\\begin{equation}\\label{kc}\nP(\\mathcal{K}^c)=\\bigo{n^{-10}}.\n\\end{equation}\nFor $1\\leq k\\leq N$, let\n$$\\mathcal{R}'_k = \\{z\\in D_{\\alpha}':d(z,I_{k}')\\leq 10c_0\\log n\\}$$\nand note that $I_k\\subset \\mathcal{R}'_k$ and for any $z\\in I_k, d(z,\\partial \\mathcal{R}'_k)\\geq 2c_0\\log n$. By Lemma \\ref{toofar} (b),\n$$\\{S_{T_{D_{\\alpha}}}\\in I_k, T_{D_{\\alpha}}\\in [j-1,j), \\mathcal{K}\\}\\subseteq \\{B_{j-1}\\in \\mathcal{R}'_k, \\mathcal{K} \\\n.$$\nThen, by \\eqref{kc}, since on $\\mathcal{K}, \\tau_{D_{\\alpha}'}\\geq T_{D_{\\alpha}}$,\n\\begin{eqnarray}\\label{main3}\\notag\nP^x(S_{T_{D_{\\alpha}}}\\in I_k) & \\leq &  \\sum_{j\\geq 1}P^x(S_{T_{D_{\\alpha}}}\\in I_k, T_{D_{\\alpha}}\\in [j-1,j), \\mathcal{K})+\\bigo{n^{-10}}\\\\ \\notag\n&\\leq & \\sum_{j\\geq 1} P^x(S_{T_{D_{\\alpha}}}\\in I_k, T_{D_{\\alpha}}\\in [j-1,j), \\tau_{D_{\\alpha}'}\\geq j-1, B_{j-1}\\in\\mathcal{R}'_k, \\mathcal{K})+\\bigo{n^{-10}\n\\\\ \\notag\n&\\lesssim & \\sum_{j\\geq 1}P^x(T_{D_{\\alpha}}\\in [j-1,j), \\tau_{D_{\\alpha}'\\setminus \\mathcal{R}'_k}\\leq \\tau_{D_{\\alpha}'})+\\bigo{n^{-10}}\\\\  \n&= & P^x(\\tau_{D'_{\\alpha}\\setminus \\mathcal{R}'_k}\\leq \\tau_{D_{\\alpha}'}\n+\\bigo{n^{-10}}\n\\end{eqnarray}\n\n \n With the convention $I'_0=I'_{N+1}=\\emptyset$, we let, for $1\\leq k\\leq N$,\n$$I_{k,+}'=\\cup_{i=k-1}^{k+1}I_i'.\n$$\n\n We now claim that \n\\begin{equation}\\label{comparable}\nP^x(\\tau_{D'_{\\alpha}\\setminus \\mathcal{R}_k}\\leq \\tau_{D_{\\alpha}'})\\lesssim P^x(B_{\\tau_{D_{\\alpha}'}}\\in I_{k,+}').\n\\end{equation}\nIndeed, by the strong Markov property for Brownian motion,\n\\begin{eqnarray*}\nP^x(\\tau_{D'_{\\alpha}\\setminus \\mathcal{R}'_k}\\leq \\tau_{D_{\\alpha}'})&\\leq & P^x(B_{\\tau_{D_{\\alpha}'}}\\in I_{k,+}')+P^x(\\tau_{D'_{\\alpha}\\setminus \\mathcal{R}'_k}\\leq \\tau_{D_{\\alpha}'},B_{\\tau_{D_{\\alpha}'}}\\not\\in I_{k,+}')\\\\\n&=&P^x(B_{\\tau_{D_{\\alpha}'}}\\in I_{k,+}')+P^x(\\tau_{D'_{\\alpha}\\setminus \\mathcal{R}'_k}\\leq \\tau_{D_{\\alpha}'})\\bigo{\\log^{-1} n}.\n\\end{eqnarray*}\nEquations \\eqref{main3} and \\eqref{comparable} now imply\n$$P^x(S_{T_{D_{\\alpha}}}\\in I_k)  \\lesssim P^x(B_{\\tau_{D_{\\alpha}'}}\\in I_{k,+}') +\\bigo{n^{-10}}$$\nand the lemma now follows from an slight modification of Lemma \\ref{beurling1} to $D_{\\alpha}'$ (note that $D_{\\alpha}'$ is not just a rescaled version of $D_{\\alpha}$, so Lemma \\ref{beurling1} cannot be applied directly but the argument of the proof of that lemma yields the same bound for $P^x(B_{\\tau_{D_{\\alpha}'}}\\in I_k')$ as in Lemma \\ref{beurling1}).\n\n\n\n\n\n\n\nIf $\\alpha = 0$, we need to use a slightly different argument from the one we just used. For a set $D\\subset \\C$, we let \n$$\\sigma_D=\\tau_{D^c}=\\inf\\{t\\geq 0: B_t\\in D\\}$$ \nbe the first hitting time of $D$ by $B$.\nWe will write $D_0$ for the set $D_\\alpha$ with $\\alpha=0$ and for $z\\in\\C, r\\in\\R_+$, we let $C(z,r)$ be the circle of radius $r$, centered at $z$. With $a$ as in the statement of the lemma, we let \n$$\nb=\\max\\{ac_0,2c_0\\}\n$$\nand define\n$$\\mathcal{S}=\\{z\\in \\C:d(z,\\bd D_0)\\leq b\\log n\\}\\setminus\\{z\\in\\C:|\\text{Im}(z+z_0)|\\leq b\\log n,|\\text{Re}(z+z_0)|\\leq b\\log n\\},$$\n$$\\S_{\\text{top}}=\\{z\\in \\bd\\S\\cap D_0:\\text{Im}(z+z_0)=b\\log n\\} \\cup C(-z_0,2n-b\\log n),$$ \n$$\\S_{\\text{bot}}=\\{z\\in \\bd\\S\\cap D_0:\\text{Im}(z+z_0)=-b\\log n\\} \\cup C(-z_0,2n+b\\log n),$$\n$$\\S_{\\text{end}}=\\{z\\in\\C:|\\text{Im}(z+z_0)|\\leq b\\log n,|\\text{Re}(z+z_0)|\\leq b\\log n\\},$$\n$$\\L_1= \\{z\\in D_0\\cup\\S:\\text{Im}(z+z_0)=0, \\text{Re}(z+z_0)\\leq b\\log n\\},$$\nFor $1\\leq k\\leq N$, let\n$$\\mathcal{S}_k=\\{z\\in \\S:d(z,\\bd D_0)=d(z,I_k)\\}$$\nand for $2\\leq k\\leq N-1$, define\n$$\\mathcal{S}_k^+=\\S_{k-1}\\cup\\S_{k}\\cup\\S_{k+1}.$$\n \\begin{figure}[htb!]\n\\centering%\n\\includegraphics[scale=0.84]{Slit.pdf}\n\\caption\nThe protagonists of the proof of Lemma \\ref{beurling2} in the case $\\alpha=0$.}\n\\label{Fig2}\n\\end{figure}\nNote that $b$ is defined in such a way that if $x$ is as in the statement of the lemma, then $x\\in\\S\\cup\\mathcal{D}$. We assume for the rest of the proof, without loss of generality, that $\\text{Im}(x+z_0)\\geq 0$. The main idea of this proof is to start $B$ and $S$ from $x$ and couple them as in \\eqref{kmt} and note that in order for $S$ to leave $D_0$ at $I_k, 2\\leq k\\leq N-1$, $B$ can't \nreach $\\S_{\\text{bot}}$ before entering $\\mathcal{S}_k^+$ or hitting $\\L_1\n$, since in that case, $S$ would necessarily either have hit $\\R_+$ at a point outside of $I_K$ or have hit the circle $C(-z_0,2n)$ and would therefore have left $D_0$ without hitting $I_k$. \n\nWe let\n$$\\eta\\!=\\! \\inf\\{t\\geq 0\\!:\\! \\exists \\, s\\leq t \\text{\u00a0s.t. } \\!\\! B_s\\in \\S_{\\text{top}}, B_t\\!\\in \\S_{\\text{bot}}, B[s,t]\\subseteq \\S\\! \\text{\u00a0or }  \\!B_s\\!\\in \\S_{\\text{bot}}, B_t\\in \\S_{\\text{top}}, B[s,t]\\subseteq \\S\\},$$\n\nThe times $\\eta$ and $\\sigma_{\\S_{\\text{bot}}}$ should typically be close to each other and it would be convenient below to be able to replace $\\eta$ by $P^x(S_{T_{D_0}}\\in I_k)$. The main difficulty in our estimate of $P^x(S_{T_{D_0}}\\in I_k)$ is dealing with the cases where $\\max\\{\\sigma_{\\S_{\\text{bot}}},\\sigma_{\\S_{\\text{top}}}\\}\\neq \\eta$. This can happen either in the vicinity of $\\S_{\\text{bot}}$ or if the Brownian path avoids $\\S_{\\text{bot}}$ but hits $\\S_{\\text{top}}$ ``from above\" and $\\S_{\\text{bot}}$ from below. \n\nNote first that\n\\begin{equation}\\label{firstdecomp} \nP^x(S_{T_{D_0}}\\in I_k)  \\leq  P^x(\\sigma_{\\mathcal{S}_k^+}\\leq \\eta) = P^x(\\sigma_{\\mathcal{S}_k^+}\\leq \\eta \\wedge \\sigma_{\\S_{\\text{end}}})+ P^x( \\sigma_{\\S_{\\text{end}}}\\leq \\sigma_{\\mathcal{S}_k^+}\\leq \\eta)\n\\end{equation}\nand that \n\\begin{equation}\\label{simplify}\n\\begin{aligned}\nP^x(\\sigma_{\\mathcal{S}_k^+}\\leq \\eta \\wedge \\sigma_{\\S_{\\text{end}}})& \\leq  P^x(\\sigma_{\\mathcal{S}_k^+}\\leq \\sigma_{\\S_{\\text{bot}}})\n+P^x(\\sigma_{\\L_1}\\leq \\sigma_{\\S_{\\text{bot}}}\\leq \\sigma_{\\mathcal{S}_k^+} \\leq \\eta\\wedge \\sigma_{\\S_{\\text{end}}})\\\\\n& \\leq  2P^x(\\sigma_{\\mathcal{S}_k^+}\\leq \\sigma_{\\S_{\\text{bot}}}),\n\\end{aligned}\n\\end{equation}\nwhere we use the reflection principle at time $\\sigma_{\\L_1}$ and the fact that for every path starting in $\\L_1$ for which $\\sigma_{\\S_{\\text{bot}}}\\leq \\sigma_{\\mathcal{S}_k^+}\\leq \\eta\\wedge \\sigma_{\\S_{\\text{end}}}$ there is a there is a reflection about $\\L_1$ for which $\\sigma_{\\mathcal{S}_k^+}\\leq \\sigma_{\\S_{\\text{bot}}}$.\n\nNote that if we define for $2\\leq k\\leq N-1$, \n$$I_{k, \\text{bot}}^{++}=\\{z\\in\\mathcal{S}_{\\text{bot}}: z\\in \\cup_{i=k-2}^{k+2}\\mathcal{S}_k\\},$$\nwith the convention $\\mathcal{S}_0=\\mathcal{S}_{N+1}=\\emptyset$, the Markov property at time $\\sigma_{\\S_{\\text{bot}}}$ gives\n\\begin{equation*}\\label{fromStoI}\n\\begin{aligned}\nP^x(\\sigma_{\\mathcal{S}_k^+}\\leq \\sigma_{\\S_{\\text{bot}}})& \\leq  P^x(\\sigma_{\\mathcal{S}_k^+}\\leq \\sigma_{\\S_{\\text{bot}}}, B(\\sigma_{\\S_{\\text{bot}}})\\in I_{k, \\text{bot}}^{++})+  P^x(\\sigma_{\\mathcal{S}_k^+}\\leq \\sigma_{\\S_{\\text{bot}}}, B(\\sigma_{\\S_{\\text{bot}}})\\not\\in I_{k, \\text{bot}}^{++})\\\\\n& \\leq  P^x(B(\\sigma_{\\S_{\\text{bot}}})\\in I_{k, \\text{bot}}^{++}) +P^x(\\sigma_{\\mathcal{S}_k^+}\\leq \\sigma_{\\S_{\\text{bot}}})\\bigo{\\frac{1}{\\log n}},\n\\end{aligned}\n\\end{equation*}\nso\n\\begin{equation}\\label{fromStoI}\nP^x(\\sigma_{\\mathcal{S}_k^+}\\leq \\sigma_{\\S_{\\text{bot}}})\\lesssim P^x(B(\\sigma_{\\S_{\\text{bot}}})\\in I_{k, \\text{bot}}^{++})\\leq\\frac{(k_0k)^{c_{\\alpha}-1}}{(k^{c_{\\alpha}}-k_0^{c_{\\alpha}})^2\\log^{c_\\alpha} n},\n\\end{equation}\nwith $c_{\\alpha}=1\/2$, where the second inequality follows from Lemma \\ref{beurling1}.\n\nEquations \\eqref{simplify} and \\eqref{fromStoI} now give a bound for the first term on the right of the equality in \\eqref{firstdecomp}: With $c_{\\alpha}=1\/2$,\n\\begin{equation}\\label{firstterm}\nP^x(\\sigma_{\\mathcal{S}_k^+}\\leq \\eta \\wedge \\sigma_{\\S_{\\text{end}}})\\lesssim \\frac{(k_0k)^{c_{\\alpha}-1}}{(k^{c_{\\alpha}}-k_0^{c_{\\alpha}})^2\\log^{c_\\alpha} n},\n\\end{equation}\n\n\n For the second term in that equality, we have\n\\begin{equation}\\label{043020}\nP^x( \\sigma_{\\S_{\\text{end}}}\\leq \\sigma_{\\mathcal{S}_k^+}\\leq \\eta)\\leq P^x( \\sigma_{\\S_{\\text{end}}}\\leq \\sigma_{\\mathcal{S}_k^+}\\wedge \\eta)\\sup_{y\\in\\S_{\\text{end}}}P^y(\\sigma_{\\mathcal{S}_k^+}\\leq \\eta).\n\\end{equation}\n\nNote that for \nthe last probability in \\eqref{043020}, we can't use the argument of \\eqref{simplify}. Instead we define, for $1\\leq j\\leq N$,\n$$I_{j,\\text{bot}}=\\{z\\in \\S_{\\text{bot}}:\\text{Re}(z+z_0)\\in [(j-1)\\log^2 n,j\\log^2 n)\\}.$$\nIf we define \n\\begin{equation}\\label{P}\nP=\\sup_{y\\in\\S_{\\text{end}}}P^y(\\sigma_{\\mathcal{S}_k^+}\\leq \\eta),\n\\end{equation}\nthen by symmetry, there is a point \n\\begin{equation*}\\label{y0}\ny_0\\in\\S_{\\text{end}}\\cap\\{z\\in D_0:\\text{Im}(z+z_0)\\geq 0\\}\n\\end{equation*}\nsuch that $P=P^{y_0}(\\sigma_{\\mathcal{S}_k^+}\\leq \\eta)$. For such a point, there is $0<C<1$ such that \n\\begin{equation}\\label{travelabit}\nP^{y_0}(B(\\sigma_{\\S_{\\text{bot}}})\\in I_{1,\\text{bot}}, \\text{Re}(z+z_0)\\leq 2b\\log n)=C.\n\\end{equation}\n\n Applying the strong Markov property at $\\sigma_{I_{j,\\text{bot}}}$ and $\\sigma_{\\S_{\\text{end}}}$, we have\n\\begin{equation}\\label{043020b}\n\\begin{aligned}\nP & \\leq P^{y_0}(\\sigma_{\\mathcal{S}_k^+}\\leq \\sigma_{\\S_{\\text{bot}}})+\\sum_{j=1}^N P^{y_0}(B(\\sigma_{\\S_{\\text{bot}}})\\in I_{j,\\text{bot}}, \\sigma_{\\S_{\\text{bot}}}\\leq \\sigma_{\\mathcal{S}_k^+}\\leq \\eta)\\\\\n& \\leq P^{y_0}(\\sigma_{\\mathcal{S}_k^+}\\leq \\sigma_{\\S_{\\text{bot}}})+\\sum_{j=1}^NP^{y_0}(B(\\sigma_{\\S_{\\text{bot}}})\\in I_{j,\\text{bot}})\\sup_{y\\in I_{j,\\text{bot}}} P^y(\\sigma_{\\mathcal{S}_k^+}\\leq \\eta\\wedge \\sigma_{\\S_{\\text{end}}})\\\\\n&\\hspace{6pc}+\\sum_{j=1}^N P^{y_0}(B(\\sigma_{\\S_{\\text{bot}}})\\in I_{j,\\text{bot}})\\sup_{y\\in I_{j,\\text{bot}}} P^y(\\sigma_{\\S_{\\text{end}}}\\leq \\sigma_{\\mathcal{S}_k^+}\\wedge \\eta)\\cdot P\n\\end{aligned}\n\\end{equation}\n\nNote that Lemma \\ref{beurling1} implies that for $j\\geq 2$,\n\\begin{equation}\\label{01-29-1}\n\\sup_{y\\in\\S_{\\text{end}}}P^y(B(\\sigma_{\\S_{\\text{bot}}})\\in I_{j,\\text{bot}})\\lesssim  \\frac{1}{j^{3\/2}\\log n}\n\\end{equation}\nand \n\\begin{equation}\\label{05-01-20}\n\\sup_{y\\in I_{j,\\text{bot}}} P^y(\\sigma_{\\mathcal{S}_k^+}\\leq \\eta\\wedge \\sigma_{\\S_{\\text{end}}}) \\lesssim \\frac{(jk)^{-1\/2}}{(k^{1\/2}-j^{1\/2})^2\\log^{1\/2} n}.\n\\end{equation}\nUsing \\eqref{travelabit}, \\eqref{01-29-1}, and \\eqref{05-01-20} in \\eqref{043020b}, we see that\n\\begin{eqnarray*}\nP &\\lesssim & P^{y_0}(\\sigma_{\\mathcal{S}_k^+}\\leq \\sigma_{\\S_{\\text{bot}}})\\\\\n&&\\hspace{2pc} +\\sup_{y\\in I_{1,\\text{bot}}} P^y(\\sigma_{\\mathcal{S}_k^+}\\leq \\eta\\wedge \\sigma_{\\S_{\\text{end}}})+\\sum_{j=2}^N\\frac{1}{j^{3\/2}\\log n}\\frac{(jk)^{-1\/2}}{(k^{1\/2}-j^{1\/2})^2\\log^{1\/2} n}\\\\\n&&\\hspace{4pc}+CP+\\sup_{\\substack{y\\in I_{1,\\text{bot}},\\\\\\text{Re}(y+z_0)\\geq 2b\\log n}} P^y(\\sigma_{\\S_{\\text{end}}}\\leq \\sigma_{\\mathcal{S}_k^+}\\wedge \\eta)P\\\\\n&&\\hspace{6pc}+\\sum_{j=2}^N\\frac{1}{j^{3\/2}\\log n}\\sup_{y\\in I_{j,\\text{bot}}} P^y(\\sigma_{\\S_{\\text{end}}}\\leq \\sigma_{\\mathcal{S}_k^+}\\wedge \\eta)\\cdot P, \n\\end{eqnarray*}\nwhich implies, since $\\sum_{j=2}^N\\frac{1}{j^{3\/2}\\log n}=o(1)$ and $\\displaystyle{\\sup_{\\substack{y\\in I_{1,\\text{bot}},\\\\\\text{Re}(y+z_0)\\geq 2b\\log n}} P^y(\\sigma_{\\S_{\\text{end}}}\\leq \\sigma_{\\mathcal{S}_k^+}\\wedge \\eta)}<1$, that \n\\begin{eqnarray*}\nP &\\lesssim & P^{y_0}(\\sigma_{\\mathcal{S}_k^+}\\leq \\sigma_{\\S_{\\text{bot}}})+\\sup_{y\\in I_{1,\\text{bot}}} P^y(\\sigma_{\\mathcal{S}_k^+}\\leq \\eta\\wedge \\sigma_{\\S_{\\text{end}}})+\\sum_{j=2}^N\\frac{1}{j^{3\/2}\\log n}\\frac{(jk)^{-1\/2}}{(k^{1\/2}-j^{1\/2})^2\\log^{1\/2} n}\\\\\n\\end{eqnarray*}\nUsing the same argument as in \\eqref{fromStoI} and the fact that $\\sum_{j=2}^N\\frac{1}{j^{2}}\\frac{1}{(k^{1\/2}-j^{1\/2})^2}\\lesssim k^{-1}$ (which can be shown, for instance, with help of the ideas in the second half of the proof of Proposition \\ref{expdiff} below), we get \n\\begin{equation}\\label{Pbound}\nP \\lesssim  \\frac{k^{-3\/2}}{\\log^{1\/2} n}+\\frac{1}{\\log^{3\/2} n}\\sum_{j=2}^N\\frac{1}{j^{2}}\\frac{1}{(k^{1\/2}-j^{1\/2})^2}\\lesssim  \\frac{k^{-3\/2}}{\\log^{1\/2} n}.\n\\end{equation}\nSome of the ideas used in the argument leading to the bound in \\eqref{Pbound} also imply\n\\begin{equation}\\label{050320}\nP^x( \\sigma_{\\S_{\\text{end}}}\\leq \\sigma_{\\mathcal{S}_k^+}\\wedge \\eta)\\lesssim  \\frac{k_0^{-3\/2}}{\\log^{1\/2} n}.\n\\end{equation}\nIt now follows from \\eqref{043020}, \\eqref{P}, \\eqref{Pbound}, and \\eqref{050320} that\n\\begin{equation}\\label{final}\nP^x( \\sigma_{\\S_{\\text{end}}}\\leq \\sigma_{\\mathcal{S}_k^+}\\leq \\eta)\\lesssim \\frac{(k_0k)^{-3\/2}}{\\log  n}. \n\\end{equation}\nSince this bound is smaller than that in \\eqref{firstterm}, combining \\eqref{firstdecomp}, \\eqref{firstterm}, and \\eqref{final} yields the lemma when $\\alpha=0$ and $2\\leq k\\leq N-1$. A similar argument can be used to handle the cases $k=1$ and $k=N$, which completes the proof in the case $\\alpha = 0$ and thus of the lemma.\n\\end{proof}\n\n\\subsection{Obtaining the Rate of Convergence for the Green's Function}\n\n\n\nThe following consequence of the lemmas of the previous section is a key element of the proof of Theorem \\ref{green}:\n\n\\begin{prop}\\label{expdiff}\nSuppose $x\\in D_{\\alpha}\\cap\\Z^2,y\\in D_{\\alpha}$ are such that $d(x,\\bd D_{\\alpha})\\leq 4c_0\\log n$, $d(x,y)\\leq 3c_0\\log n$, and a subset of $\\bd D_{\\alpha}$ closest to $x$ is $I_{k_0}$. Then \n \n $$E^{x,y}\\left[\\bigg|\\log \\frac{|S_{T_{D_{\\alpha}}}|}{|B_{\\tau_{D_{\\alpha}}}|}\\bigg|\\right]\\lesssim k_0^{c_\\alpha-1}n^{-c_{\\alpha}}\\log^{c_{\\alpha}+1} n,$$\nwhere \n$$c_{\\alpha}=\\frac{1}{2}+\\frac{\\alpha}{4\\pi-2\\alpha}.$$\n\\end{prop}\n\n\\begin{proof} If for $1\\leq k, \\ell \\leq N, z\\in I_k, w\\in I_\\ell$, then \n$$|\\log(|z|\/|w|)|\\leq \\left|\\log\\left(\\frac{|w|+|z-w|}{|w|}\\right)\\right|=\\bigo{\\frac{(|k-\\ell|+1)\\log^2 n}{n}},$$\nso\n\\begin{eqnarray}\\label{last1}\\notag\nE^{x,y}\\left[\\bigg|\\log \\frac{|S_{T_{D_{\\alpha}}}|}{|B_{\\tau_{D_{\\alpha}}}|}\\bigg|\\right]\n& \\lesssim & \\frac{\\log^2 n}{n}\\sum_{k,\\ell=1}^N P^x(S_{T_{D_{\\alpha}}}\\in I_k)P^y(B_{\\tau_{D_{\\alpha}}}\\in I_\\ell)(|k-k_0|+|\\ell-k_0|+1)\\\\ \\notag\n& \\lesssim & \\frac{\\log^2 n}{n}\\left(1+\\sum_{k=1}^N P^x(S_{T_{D_{\\alpha}}}\\in I_k)|k-k_0|+\\sum_{\\ell=1}^N P^y(B_{\\tau_{D_{\\alpha}}}\\in I_\\ell)|\\ell-k_0|\\right)\\\\\n&\\leq &\\frac{\\log^2 n}{n}+\\sum_{k=1}^{N}\\frac{(k_0 k)^{c_\\alpha-1}}{(k_0^{c_\\alpha}-k^{c_\\alpha})^2\\log^{c_\\alpha} n}\\frac{|k_0-k|\\log^2 n}{n},\n\\end{eqnarray}\nwhere the last inequality is a consequence of Lemmas \\ref{beurling1} and  \\ref{beurling2}.\nIf $N\\geq \\lceil 3k_0\/2\\rceil$, then\n\\begin{equation}\\label{last2}\n\\sum_{\\substack{k =1\\\\ k\\neq k_0}}^N\\frac{k^{c_\\alpha - 1}|k_0-k|}{(k_0^{c_\\alpha}-k^{c_\\alpha})^2}=S_1+S_2+S_3+S_4,\n\\end{equation}\nwhere\n$$S_1=\\sum_{k\\leq \\lfloor k_0\/2\\rfloor}\\frac{k^{c_\\alpha - 1}(k_0-k)}{(k_0^{c_\\alpha}-k^{c_\\alpha})^2}, \\qquad S_2=\\sum_{k=\\lceil k_0\/2\\rceil}^{k_0-1}\\frac{k^{c_\\alpha - 1}(k_0-k)}{(k_0^{c_\\alpha}-k^{c_\\alpha})^2},$$\n$$S_3=\\sum_{k=k_0+1}^{\\lfloor 3k_0\/2\\rfloor}\\frac{k^{c_\\alpha - 1}(k-k_0)}{(k^{c_\\alpha}-k_0^{c_\\alpha})^2}, \\qquad S_4=\\sum_{k=\\lceil 3k_0\/2\\rceil}^{N}\\frac{k^{c_\\alpha - 1}(k-k_0)}{(k^{c_\\alpha}-k_0^{c_\\alpha})^2}.$$\nClearly, $S_1\\lesssim k_0^{1-c_{\\alpha}}$. It is easy to see that for $\\lceil k_0\/2\\rceil \\leq k \\leq k_0-1$, \n$$\\frac{k_0-k}{(k_0^{c_\\alpha}-k^{c_\\alpha})^2}\\lesssim \\frac{k_0^{2-2c_\\alpha}}{k_0-k},$$\nso \n$$S_2\\lesssim \\sum_{j=1}^{\\lfloor k_0\/2\\rfloor}\\frac{(k_0-j)^{c_{\\alpha}-1}k_0^{2-2c_\\alpha}}{j}\\lesssim k_0^{1-c_\\alpha}\\log (k_0).$$\nSimilarly, $S_3\\lesssim k_0^{1-c_\\alpha}\\log (k_0)$. Finally, using the fact that if $k\\geq \\lceil 3k_0\/2\\rceil$, there is $C>0$ such that $(1-(k_0\/k)^{c_\\alpha})\\geq C$, we see that \n$$S_4\\lesssim \\sum_{k=\\lceil 3k_0\/2\\rceil}^{N}k^{-c_\\alpha}\\lesssim N^{1-c_\\alpha}-(3k_0\/2)^{1-c_\\alpha}.$$\nCombining the bounds for $S_1, S_2, S_3, S_4$ with \\eqref{last2} and \\eqref{last1}, we get, in the case $N\\geq \\lceil 3k_0\/2\\rceil$,\n$$\\sum_{\\substack{k =1\\\\ k\\neq k_0}}^N\\frac{k^{c_\\alpha - 1}|k_0-k|}{(k_0^{c_\\alpha}-k^{c_\\alpha})^2}\\lesssim N^{1-c_\\alpha} \\log N,$$\nso \n$$E^{x,y}\\left[\\bigg|\\log \\frac{|S_{T_{D_{\\alpha}}}|}{|B_{\\tau_{D_{\\alpha}}}|}\\bigg|\\right]\\lesssim \\frac{k_0^{c_\\alpha -1}\\log^{2-c_\\alpha} n}{n}N^{1-c_\\alpha} \\log N\\lesssim \\frac{k_0^{c_\\alpha -1}\\log{2-c_\\alpha} n}{n}\\left(\\frac{n}{\\log^2 n}\\right)^{1-c_\\alpha} \\log n\n$$\nwhich proves the proposition in the case where $N\\geq \\lceil 3k_0\/2\\rceil$. If $N< \\lceil 3k_0\/2\\rceil$, $S_4=0$ and the bounds for $S_1, S_2, S_3$ are the same.\n\\end{proof}\n\n\n\n\n\n\n\n\nThe strategy we use in the proof of Theorem \\ref{green} is similar to that in \\cite{kl} and \\cite{bjk}, though we handle some technical issues slightly differently in the present paper. We couple $B$ and $S$ until they are close to $\\bdD_{\\alpha}$ but are likely not to have left $D_{\\alpha}$ yet. We then let each run independently and use Proposition \\ref{expdiff}. The technical difficulty stems in the fact that we would like to use the strong Markov property but $B$ and $S$ are not strong Markov when considered jointly under the coupling.\n\n\\begin{proof} [Proof of Theorem \\ref{green}]\n\nIn light of \\eqref{discretegreenrep}, \\eqref{ax}, and \\eqref{contgreenrep} above,  since there exist $c_1, c_2>0$ such that \n\\begin{equation}\\label{inrad}\nc_1n\\leq \\inf_{z \\in D_{\\alpha}}|z|\\leq \\sup_{z \\in D_{\\alpha}}|z|\\leq c_2n,\n\\end{equation}\nwe have\n\\begin{equation}\\label{mainproof1}\n\\left|G_{D_{\\alpha}}(w)-\\frac{2}{\\pi}g_{D_{\\alpha}}(w)\\right|=E^w\\left[\\bigg|\\log \\frac{|S_{T_{D_{\\alpha}}}|}{|B_{\\tau_{D_{\\alpha}}}|}\\bigg|\\right]+\\bigo{|w|^{-2}}.\n\\end{equation}\nIn order to prove the theorem, we just need to show that the upper bound of Proposition \\ref{expdiff} with $k_0^{c_{\\alpha}-1}$ replaced by the obvious upper bound of 1 also holds for the expected value in \\eqref{mainproof1}.  We let $\\mathcal{K}$ be as in \\eqref{k}. Note that \\eqref{inrad} implies that \n$$\\bigg|\\log \\frac{|S_{T_{D_{\\alpha}}}|}{|B_{\\tau_{D_{\\alpha}}}|}\\bigg|\\asymp 1,$$\nso\n$$E^w\\left[\\bigg|\\log \\frac{|S_{T_{D_{\\alpha}}}|}{|B_{\\tau_{D_{\\alpha}}}|}\\bigg|\\right] \\leq E^w\\left[\\bigg|\\log \\frac{|S_{T_{D_{\\alpha}}}|}{|B_{\\tau_{D_{\\alpha}}}|}\\bigg|; \\mathcal{K}\\right]+ \\bigo{n^{-10}}$$\n\n\n\n\n\nFor the rest of the proof, we let $c_0$ be as in \\eqref{k}\nand define \n$$\\eta= \\inf\\{t\\geq 0: d(B_t,\\partial D_{\\alpha})\\leq 2c_0\\log n\\}$$\n(see Figure \\ref{Fig1}) and note that on the event $\\mathcal{K}$, $\\eta\\leq \\tau_{D_\\alpha}\\wedge T_{D_\\alpha}$.\nThen \n\\begin{eqnarray*}\nE^w\\left[\\bigg|\\log \\frac{|S_{T_{D_{\\alpha}}}|}{|B_{\\tau_{D_{\\alpha}}}|}\\bigg|; \\mathcal{K}\\right] &=& \\sum_{k\\geq 0} E^w\\left[\\bigg|\\log \\frac{|S_{T_{D_{\\alpha}}}|}{|B_{\\tau_{D_{\\alpha}}}|}\\bigg|; \\mathcal{K}; \\eta\\in [k,k+1)\\right]\\\\\n& \\leq & \\sum_{k\\geq 0} E^w\\left[\\bigg|\\log \\frac{|S_{T_{D_{\\alpha}}}|}{|B_{\\tau_{D_{\\alpha}}}|}\\bigg|; \\mathcal{E}\\right],\n\\end{eqnarray*}\nwhere $\\mathcal{E}=\\{\\max\\{d(S_k, \\bd D_{\\alpha}), d(B_k, \\bd D_{\\alpha})\\}\\leq 4c_0\\log n; d(B_k,S_k)\\leq 3c_0\\log n; k\\leq \\tau_{D_\\alpha}\\wedge T_{D_\\alpha}\\}$ and the last inequality follows from Lemma \\ref{toofar}. The proof of the theorem is now complete if we apply the Markov property at time $k$ and use Proposition \\ref{expdiff}.\n\n\n\n\n\\end{proof}\n\n\\section{Acknowledgments}\nThe author gratefully acknowledges support through PSC-CUNY Award \\# 61514-00 49.\n\n\\bibliographystyle{plain}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}