Conservation laws of random matrix theory

Summary

This paper presents an overview of the derivation and significance of recently derived conservation laws for the matrix moments of Hermitian random matrices with dominant exponential weights that may be either even or odd. This is based on a detailed asymptotic analysis of the partition function for these unitary ensembles and their scaling limits. As a particular application we derive closed form expressions for the coefficients of the genus expansion for the associated free energy in a particular class of dominant even weights. These coefficients are generating functions for enumerating g-maps, related to graphical combinatorics on Riemann surfaces. This generalizes and resolves a 30+ year old conjecture in the physics literature related to quantum gravity.

1. Introduction

We present an overview of some recent developments in the application of random matrix analysis to the topological combinatorics of surfaces. Such applications have a long history about which we should say a few words at the outset. The combinatorial objects of interest here are maps. A map is an embedding of a graph into a compact, oriented and connected surface X with the requirement that the complement of the graph in X should be a disjoint union of simply connected open sets. If the genus of X is g, this object is referred to as a g-map. The notion of g-maps was introduced by Tutte [1968] and his collaborators in the 1960s as part of their investigations of the four color conjecture.

In the early 1980s Bessis, Itzykson and Zuber, a group of physicists studying 't Hooft's diagrammatic approaches to large-N expansions in quantum field theory, discovered a profound connection between the problem of enumerating g-maps and random matrix theory [Bessis et al. 1980]. That seminal work was the basis for bringing asymptotic analytical methods into the study of maps and other related combinatorial problems. Subsequently other physicists [Douglas and Shenker 1990; Gross and Migdal 1990] realized that the matrix model diagrammatics described by Bessis et al. provide a natural means for discretizing the Einstein–Hilbert action in two dimensions.