Asymptotics of the partition function for random matrices via Riemann-Hilbert techniques, and applications to graphical enumeration

TL;DR

This paper develops a rigorous asymptotic expansion for the partition function in random matrix theory using Riemann-Hilbert techniques, linking it to graphical enumeration and eigenvalue density analysis.

Contribution

It provides the first complete proof of a large N expansion for the partition function for general measures, confirming conjectures and establishing analytic properties of expansion coefficients.

Findings

Established a large N expansion for the partition function.

Proved coefficients are generating functions for map enumeration.

Derived a uniform large N eigenvalue density expansion.

Abstract

We study the partition function from random matrix theory using a well known connection to orthogonal polynomials, and a recently developed Riemann-Hilbert approach to the computation of detailed asymptotics for these orthogonal polynomials. We obtain the first proof of a complete large N expansion for the partition function, for a general class of probability measures on matrices, originally conjectured by Bessis, Itzykson, and Zuber. We prove that the coefficients in the asymptotic expansion are analytic functions of parameters in the original probability measure, and that they are generating functions for the enumeration of labelled maps according to genus and valence. Central to the analysis is a large N expansion for the mean density of eigenvalues, uniformly valid on the entire real axis.