Large N expansions and Painlev\'e hierarchies in the Hermitian matrix model

TL;DR

This paper develops a method to analyze the large N asymptotics of Hermitian matrix models, linking them to Painlevé hierarchies through continuum limits of the string equation and Toda hierarchy resolvents.

Contribution

It introduces a novel approach to compute asymptotics of matrix models using Lax operators and explicitly relates critical behaviors to Painlevé hierarchies.

Findings

Explicit formulas for asymptotics in one-cut and two-cut models.

Connection established between matrix model critical points and Painlevé hierarchies.

Method applicable to general even potentials.

Abstract

We present a method to characterize and compute the large N formal asymptotics of regular and critical Hermitian matrix models with general even potentials in the one-cut and two-cut cases. Our analysis is based on a method to solve continuum limits of the discrete string equation which uses the resolvent of the Lax operator of the underlying Toda hierarchy. This method also leads to an explicit formulation, in terms of coupling constants and critical parameters, of the members of the Painlev\'e I and Painlev\'e II hierarchies associated with one-cut and two-cut critical models respectively.