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78
Asymptotic analysis of a hermitian matrix integral
 International Journal of Mathematics
, 1995
"... Abstract. The asymptotic expansion of a Hermitian matrix integral known as the Penner model is rigorously calculated. 1. Introduction. The purpose of this paper is to establish an asymptotic analysis of a Hermitian matrix integral known as the Penner model, and to calculate its asymptotic expansion. ..."
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Cited by 14 (12 self)
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Abstract. The asymptotic expansion of a Hermitian matrix integral known as the Penner model is rigorously calculated. 1. Introduction. The purpose of this paper is to establish an asymptotic analysis of a Hermitian matrix integral known as the Penner model, and to calculate its asymptotic expansion. It was proved by Penner [7] that this asymptotic series gives the orbifold Euler characteristic of the moduli spaces of pointed algebraic curves. The formula
Asymptotics of the partition function of a random matrix model
 Ann. Inst. Fourier (Grenoble
"... Dedicated to Pierre van Moerbeke on his sixtieth birthday. Abstract. We prove a number of results concerning the large N asymptotics of the free energy of a random matrix model with a polynomial potential V (z). Our approach is based on a deformation τtV (z) of V (z) to z 2, 0 ≤ t < ∞ and on the use ..."
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Cited by 14 (2 self)
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Dedicated to Pierre van Moerbeke on his sixtieth birthday. Abstract. We prove a number of results concerning the large N asymptotics of the free energy of a random matrix model with a polynomial potential V (z). Our approach is based on a deformation τtV (z) of V (z) to z 2, 0 ≤ t < ∞ and on the use of the underlying integrable structures of the matrix model. The main results include (1) the existence of a full asymptotic expansion in powers of N −2 of the recurrence coefficients of the related orthogonal polynomials, for a onecut regular V; (2) the existence of a full asymptotic expansion in powers of N −2 of the free energy, for a V, which admits a onecut regular deformation τtV; (3) the analyticity of the coefficients of the asymptotic expansions of the recurrence coefficients and the free energy, with respect to the coefficients of V; (4) the onesided analyticity of the recurrent coefficients and the free energy for a onecut singular V; (5) the double scaling asymptotics of the free energy for a singular quartic polynomial V.
On an integral representation for the genus series for 2cell embeddings
 Trans. Amer. Math. Soc
, 1994
"... Abstract. An integral representation for the genus series for maps on oriented surfaces is derived from the combinatorial axiomatisation of 2cell embeddings in orientable surfaces. It is used to derive an explicit expression for the genus series for dipoles. The approach can be extended to vertexr ..."
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Cited by 13 (6 self)
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Abstract. An integral representation for the genus series for maps on oriented surfaces is derived from the combinatorial axiomatisation of 2cell embeddings in orientable surfaces. It is used to derive an explicit expression for the genus series for dipoles. The approach can be extended to vertexregular maps in general and, in this way, may shed light on the genus series for quadrangulations. The integral representation is used in conjunction with an approach through the group algebra, C<Sn, of the symmetric group [11] to obtain a factorisation of certain Gaussian integrals. 1. A POWER SERIES REPRESENTATION FOR THE GENUS SERIES A map is a 2cell embedding of a connected unlabelled graph &, with loops and multiple edges allowed, in a closed surface 1, without boundary, which is assumed throughout to be oriented. The deletion of 9 separates 1 into regions homeomorphic to open discs, called the faces of the map, and the number of edges bordering a face is called its degree. A map is rooted by distinguishing a mutually incident vertex, edge and face. The genus series for a class of maps is
Freud's Equations For Orthogonal Polynomials As Discrete Painlevé Equations.
"... . We consider orthogonal polynomials pn with respect to an exponential weight function w(x) = exp(\GammaP (x)). The related equations for the recurrence coefficients have been explored by many people, starting essentially with Laguerre [49], in order to study special continued fractions, recurrence ..."
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Cited by 10 (0 self)
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. We consider orthogonal polynomials pn with respect to an exponential weight function w(x) = exp(\GammaP (x)). The related equations for the recurrence coefficients have been explored by many people, starting essentially with Laguerre [49], in order to study special continued fractions, recurrence relations, and various asymptotic expansions (G. Freud's contribution [28, 56]). Most striking example is n = 2twn + wn (wn+1 + wn + wn\Gamma1 ) for the recurrence coefficients pn+1 = xpn \Gamma wn pn\Gamma1 of the orthogonal polynomials related to the weight w(x) = exp(\Gamma4(tx 2 + x 4 )) (notations of [26, pp.34 36]). This example appears in practically all the references below. The connection with discrete Painlev'e equations is described here. 1. Construction of orthogonal polynomials recurrence coefficients. Consider the set fp n g 1 0 of orthonormal polynomials with respect to a weight function w on (a part of) the real line: Z 1 \Gamma1 p n (x)p m (x)w(x) dx = ffi n;m ; n...
Lectures on the asymptotic expansion of a hermitian matrix integral
 in Supersymmetry and Integrable Models, Henrik Aratin et al., Editors, Springer Lecture Notes in Physics 502
, 1998
"... Abstract. In these lectures three different methods of computing the asymptotic expansion of a Hermitian matrix integral is presented. The first one is a combinatorial method using Feynman diagrams. This leads us to the generating function of the reciprocal of the order of the automorphism group of ..."
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Cited by 10 (7 self)
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Abstract. In these lectures three different methods of computing the asymptotic expansion of a Hermitian matrix integral is presented. The first one is a combinatorial method using Feynman diagrams. This leads us to the generating function of the reciprocal of the order of the automorphism group of a tiling of a Riemann surface. The second method is based on the classical analysis of orthogonal polynomials. A rigorous asymptotic method is established, and a special case of the matrix integral is computed in terms of the Riemann ζfunction. The third method is derived from a formula for the τfunction solution to the KP equations. This method leads us to a new class of solutions of the KP equations that are transcendental, in the sense that theycannot be obtained bythe celebrated Krichever construction and its generalizations based on algebraic geometryof vector bundles on Riemann surfaces. In each case a mathematicallyrigorous wayof dealing with asymptotic series in an infinite number of variables is established. Contents
On the Counting of Colored Tangles
 Journal of Knot Theory and its Ramifications 9 (2000) 1127–1141 (preprint mathph/0002020
"... The connection between matrix integrals and links is used to define matrix models which count alternating tangles in which each closed loop is weighted with a factor n, i.e. may be regarded as decorated with n possible colors. For n = 2, the corresponding matrix integral is that recently solved in t ..."
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Cited by 9 (6 self)
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The connection between matrix integrals and links is used to define matrix models which count alternating tangles in which each closed loop is weighted with a factor n, i.e. may be regarded as decorated with n possible colors. For n = 2, the corresponding matrix integral is that recently solved in the study of the random lattice sixvertex model. The generating function of alternating 2color tangles is provided in terms of elliptic functions, expanded to 16th order (16 crossings) and its asymptotic behavior is given. 02/2000
Duality of orthogonal and symplectic matrix integrals and quaternionic feynman graphs
 Commun. Math. Phys
"... ABSTRACT. We present an asymptotic expansion for quaternionic selfadjoint matrix integrals. The Feynman diagrams appearing in the expansion are ordinary ribbon graphs and their nonorientable counterparts. We show that the 2N × 2N Gaussian Orthogonal Ensemble (GOE) and N × N Gaussian Symplectic Ense ..."
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Cited by 8 (0 self)
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ABSTRACT. We present an asymptotic expansion for quaternionic selfadjoint matrix integrals. The Feynman diagrams appearing in the expansion are ordinary ribbon graphs and their nonorientable counterparts. We show that the 2N × 2N Gaussian Orthogonal Ensemble (GOE) and N × N Gaussian Symplectic Ensemble (GSE) have exactly the same expansion term by term, except that the contributions from graphs on a nonorientable surface with odd Euler characteristic carry the opposite sign. As an application, we give a new topological proof of the known duality for correlations of characteristic polynomials, demonstrating that this duality is equivalent to Poincaré duality of graphs drawn on a compact surface. Another consequence of our graphical expansion formula is a simple and simultaneous (re)derivation of the Central Limit Theorem for GOE, GUE (Gaussian Unitary Ensemble) and GSE: The three cases have exactly the same graphical limiting formula except for an overall
Random Matrices, Graphical Enumeration and the Continuum Limit of Toda Lattices
"... Abstract. In this paper we derive a hierarchy of differential equations which uniquely determine the coefficients in the asymptotic expansion, for large N, of the logarithm of the partition function of N × N Hermitian random matrices. These coefficients are generating functions for graphical enumera ..."
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Cited by 7 (3 self)
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Abstract. In this paper we derive a hierarchy of differential equations which uniquely determine the coefficients in the asymptotic expansion, for large N, of the logarithm of the partition function of N × N Hermitian random matrices. These coefficients are generating functions for graphical enumeration on Riemann surfaces. The case that we particularly consider is for an underlying measure that differs from the Gaussian weight by a single monomial term of degree 2ν. The coupling parameter for this term plays the role of the independent dynamical variable in the differential equations. From these equations one may deduce functional analytic characterizations of the coefficients in the asymptotic expansion. Moreover, this ode system can be solved recursively to explicitly construct these coefficients as functions of the coupling parameter. This analysis of the fine structure of the asymptotic coefficients can be extended to multiple coupling parameters and we present a limited illustration of this for the case of two parameters. 1. Motivation
Noncommutative matrix integrals and representation varieties of surface groups in a finite group, Annales de l’Institut Fourier 55
, 2005
"... Abstract. A graphical expansion formula for noncommutative matrix integrals with values in a finitedimensional real or complex von Neumann algebra is obtained in terms of ribbon graphs and their nonorientable counterpart called Möbius graphs. The contribution of each graph is an invariant of the ..."
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Cited by 6 (2 self)
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Abstract. A graphical expansion formula for noncommutative matrix integrals with values in a finitedimensional real or complex von Neumann algebra is obtained in terms of ribbon graphs and their nonorientable counterpart called Möbius graphs. The contribution of each graph is an invariant of the topological type of the surface on which the graph is drawn. As an example, we calculate the integral on the group algebra of a finite group. We show that the integral is a generating function of the number of homomorphisms from the fundamental group of an arbitrary closed surface into the finite group. The graphical expansion formula yields a new proof of the classical theorems of Frobenius, Schur and Mednykh on these numbers. The purpose of this paper is to establish Feynman diagram expansion formulas for noncommutative matrix integrals over a finitedimensional real or complex von Neumann algebra. An interesting case is the real or complex group algebra of a finite group. Using the graphical expansion formulas, we give a new proof of the classical formulas for the number
Sums over graphs and integration over discrete groupoids
 Applied Categorical Structures
"... Abstract. We show that sums over graphs such as appear in the theory of Feynman diagrams can be seen as integrals over discrete groupoids. From this point of view, basic combinatorial formulas of the theory of Feynman diagrams can be interpreted as pullback or pushforward formulas for integrals ov ..."
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Cited by 5 (1 self)
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Abstract. We show that sums over graphs such as appear in the theory of Feynman diagrams can be seen as integrals over discrete groupoids. From this point of view, basic combinatorial formulas of the theory of Feynman diagrams can be interpreted as pullback or pushforward formulas for integrals over suitable groupoids.