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.

Abstract. It is well-known that the partition function of the unitary ensembles of random matrices is given by a τ -function of the Toda lattice hierarchy and those of the orthogonal and symplectic ensembles are τ -functions of the Pfaff lattice hierarchy. In these cases the asymptotic expansions of the free energies given by the logarithm of the partition functions lead to the dispersionless (i.e. continuous) limits for the Toda and Pfaff lattice hierarchies. There is a universality between all three ensembles of random matrices, one consequence of which is that the leading orders of the free energy for large matrices agree. In this paper, this universality, in the case of Gaussian ensembles, is explicitly demonstrated by computing the leading orders of the free energies in the expansions. We also show that the free energy as the solution of the dispersionless Toda lattice hierarchy gives a solution of the dispersionless Pfaff lattice hierarchy, which implies that this universality holds in general for the leading orders of the unitary, orthogonal, and symplectic ensembles.We also find an explicit formula for the two point function Fnm which represents the number of connected ribbon graphs with two vertices of degrees n and m on a sphere. The derivation is based on the Faber polynomials defined on the spectral curve of the dispersionless Toda lattice hierarchy, and 1 nm Fnm are the Grunsky coefficients of the Faber polynomials.

Random Hermitian matrices with a source term arise, for instance, in the study of non-intersecting Brownian walkers [1,20] and sample covariance matrices [4]. We consider the case when the n × n external source matrix has two distinct real eigenvalues: a with multiplicity r and zero with multiplicity n − r. The source is small in the sense that r is finite or r = O(n γ ), for 0 < γ < 1. For a Gaussian potential, Péché [29] showed that for |a| sufficiently small (the subcritical regime) the external source has no leading-order effect on the eigenvalues, while for |a| sufficiently large (the supercritical regime) r eigenvalues exit the bulk of the spectrum and behave as the eigenvalues of r × r Gaussian unitary ensemble (GUE). We establish the universality of these results for a general class of analytic potentials in the supercritical and subcritical regimes.

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The (semi-infinite) Pfaff lattice was introduced by Adler and van Moerbeke [2] to describe the partition functions for the random matrix models of GOE and GSE type. The partition functions of those matrix models are given by the Pfaffians of certain skew-symmetric matrices called the moment matrices, and they are the τ -functions of the Pfaff lattice. In this paper, we study a finite version of the Pfaff lattice equation as a Hamiltonian system. In particular, we prove the complete integrability in the sense of Arnold-Liouville, and using a moment map, we describe the real isospectral varieties of the Pfaff lattice. The image of the moment map is a convex polytope whose vertices are identified as the fixed points of the flow generated by the Pfaff lattice. 14 3. Matrix factorization and the τ -functions 16 3.1. Moment matrix and the τ -functions 17 3.2. Foliation of the phase space by F r,k (L) 21 3.3. Examples from the matrix models 22 4. Real solutions 23 4.1. Skew-orthogonal polynomials 23 4.2. Fixed points of the Pfaff flows 26 4.3. The GSE-Pfaff lattice 29 5. Geometry of the isospectral variety 31 5.1. Geometric structure of the τ -functions 31 5.2. Moment polytope 32 5.3. The moment polytope for the GSE model 35 References 36