Painlev\'e kernels in Hermitian matrix models

TL;DR

This paper explores the asymptotic behavior of Hermitian matrix models, focusing on a Painlevé II related kernel in the two matrix model with quartic potential, and discusses open problems in the field.

Contribution

It introduces a new limiting kernel for the two matrix model with quartic potential, constructed via a Riemann-Hilbert problem related to Painlevé II, advancing understanding of matrix model asymptotics.

Findings

Identification of a Painlevé II related kernel in the quartic/quadratic case

Construction of the kernel using a 4x4 Riemann-Hilbert problem

Presentation of an open problem in the asymptotic analysis of matrix models

Abstract

After reviewing the Hermitian one matrix model, we will give a brief introduction to the Hermitian two matrix model and present a summary of some recent results on the asymptotic behavior of the two matrix model with a quartic potential. In particular, we will discuss a limiting kernel in the quartic/quadratic case that is constructed out of a $4\times 4$ Riemann-Hilbert problem related to Painlev\'e II equation. Also an open problem will be presented.