Abstract. We derive the double scaling limit of eigenvalue correlations in the random matrix model at critical points and we relate it to a nonlinear hierarchy of ordinary differential equations. 1.

We show that under reasonably general assumptions, the first order asymptotics of the free energy of matrix models are generating functions for colored planar maps. This is based on the fact that solutions of the differential Schwinger-Dyson equations are, by nature, generating functions for enumerating planar maps, a remark which bypasses the use of Gaussian calculus.

Perturbation of the GUE are known in physics to be related to enumeration of graphs on surfaces. Following [7] and [8], we investigate this idea and show that for a small convex perturbation, we can perform a genus expansion: the moments of the empirical measure can be developed into a series whose g-th term is a generating function of graphs on a surface of genus g.

. We discuss a method of the asymptotic computation of moments of the normalized eigenvalue counting measure of random matrices of large order. The method is based on the resolvent identity and on some formulas relating expectations of certain matrix functions and the expectations including their derivatives or, equivalently, on some simple formulas of the perturbation theory. In the framework of this unique approach we obtain functional equations for the Stieltjes transforms of the limiting normalized eigenvalue counting measure and the bounds for the rate of convergence for the majority known random matrix ensembles. 1. Introduction Random matrix theory is actively developing. Among numerous topics of the theory and its various applications those related to the asymptotic eigenvalue distribution of random matrices of large order are of considerable interest. An important role in this branch of the theory plays the eigenvalue counting measure defined for any Hermitian or real symmetr...

Abstract. Bauer and Itzykson showed that associated to each labeled map embedded on an oriented Riemann surface there was a group generated by a pair of permutations. From this result an algorithm may be constructed for enumerating labeled maps, and this construction is easily augmented to bin the numbers by the genus of the surface the map is embedded in. The results agree with the calculations of Harer and Zagier of 1-vertex maps; with those of Bessis, Itzykson, and Zuber of 4-valent maps; and with those of Ercolani, McLaughlin, and Pierce for 2ν-valent maps. We then modify this algorithm to one which counts unoriented maps or Mobius graphs. The results in this case agree with the calculation of Goulden and Jackson on 1-vertex unoriented maps. 1.

Enumerating bipartite (with black and white vertices) planar maps according to the degree distribution of the vertices is useful to physicists. We first exhibit a bijection between these maps and some family of trees. The generating functions of these trees are then obtained with classical decomposition on the combinatorial structure of the trees. The physicists need to add Ising or hard particle models to planar maps to model particle location or spin. We can relate bijectively these maps with an additional structure to the bipartite maps. We finally enumerate the Ising and hard particle configurations on maps. (Joint work with Mireille Bousquet-Mélou from Labri) 1.

We study several-matrix models and show that when the potential is convex and a small perturbation of the Gaussian potential, the first order correction to the free energy can be expressed as a generating function for the enumeration of maps of genus one. In order to do that, we prove a central limit theorem for traces of words of the weakly interacting random matrices defined by these matrix models and show that the variance is a generating function for the number of planar maps with two vertices with prescribed colored edges. 1

Contents 1 Introduction 1 2 String Theory 1 2.1 Action Principles of Classical Strings . . . . . . . . . . . . . . . . . . . . . . . . . 1 2.2 Quantization of the Closed String . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 3 Two-Dimensional Topological Field Theories and Topological Sigma Models 9 3.1 D ! 1 Field Theory and One Matrix Models . . . . . . . . . . . . . . . . . . . . . 9 3.2 Topological Field Theory in Two Dimensions . . . . . . . . . . . . . . . . . . . . . 13 3.3 The Moduli Space of Curves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 3.4 Correlation Functions and Intersection Theory . . . . . . . . . . . . . . . . . . . . 16 3.5 Topological Sigma Models and Quantum Cohomology . . . . . . . . . . . . . . . . 18 4 Enumerative Geometry via Torus Actions 19 4.1 Torus actions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 4.2 Application of Torus Actions

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Abstract. We summarize some combinatoric problems solved by the higher Catalan numbers. These problems are generalizations of the combinatoric problems solved by the Catalan numbers. The generating function of the higher Catalan numbers appeared recently as an auxiliary function in enumerating maps and explicit computations of the asymptotic expansion of the partition function of random matrices in the unitary ensemble case. We give combinatoric proofs of the formulas for the number of genus 0 and genus 1 maps. 1. Higher Catalan Numbers The Catalan numbers solve a number of classical combinatoric problems such as the “Euler Polygon Division Problem”: how many ways are there to divide a marked polygon with j + 2 sides into triangles using edges and diagonals [3, 7, 8, 12, 16] (see figure 1). Figure 1. A polygon with 4 + 2 = 6 sides divided into 4 triangles using edges and diagonals They count the number of right-left paths along a 1-Dimensional integer lattice which stay to the right of 0 and go from 0 to 0 in 2j steps; equivalently they count Dyck paths from (0, 0) to (2j, 0) [1, 4, 15, 18]. They count the number of ways for 2j customers to line up, with j customers having only a 2-dollar bill and j customers having only a 1-dollar bill, to purchase 1-dollar widgets, so that each customer receives exact change. They count the number of non-crossing handshakes possible across a round table between n people [5]. In this paper we will explore a generalization of the Catalan numbers, the higher Catalan numbers. We will show that this generalization solves enumerative problems that are natural generalizations of the problems solved by the Catalan numbers. We will then highlight their appearance in recent results on map enumeration problems.