Random Matrices, Graphical Enumeration and the Continuum Limit of Toda Lattices

TL;DR

This paper develops a hierarchy of differential equations to determine asymptotic expansion coefficients of the logarithm of the partition function for large Hermitian random matrices, linking them to graphical enumeration on Riemann surfaces.

Contribution

It introduces a recursive method to explicitly compute asymptotic coefficients as functions of coupling parameters, extending to multiple parameters.

Findings

Derived differential equations for asymptotic coefficients.

Explicit recursive formulas for coefficients as functions of coupling.

Extended analysis to multiple coupling parameters.

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 $N \times N$ 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 $2\nu$. 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.