Combinatorial aspects of matrix models

TL;DR

This paper demonstrates that the first order asymptotics of matrix model free energy serve as generating functions for colored planar maps, linking Schwinger-Dyson equations to combinatorial enumeration without Gaussian calculus.

Contribution

It establishes a direct connection between matrix model asymptotics and combinatorial map enumeration through Schwinger-Dyson equations.

Findings

First order asymptotics correspond to generating functions for colored planar maps.

Schwinger-Dyson equations naturally encode enumeration of planar maps.

The approach bypasses Gaussian calculus in deriving combinatorial results.

Abstract

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 Schwinger-Dyson equations are, by nature, generating functions for enumerating planar maps, a remark which bypasses the use of Gaussian calculus.