Random matrix ensembles with singularities and a hierarchy of Painlev\'e III equations

TL;DR

This paper investigates the asymptotic behavior of singularly perturbed unitary invariant random matrix ensembles, revealing a hierarchy of Painlevé III equations that describe their critical phenomena.

Contribution

It introduces a hierarchy of higher order Painlevé III equations to describe asymptotics of ensembles with singular potentials, extending previous understanding of such models.

Findings

Asymptotics for partition functions of perturbed Laguerre and Gaussian ensembles.

Double scaling limits connect to a hierarchy of Painlevé III equations.

Correlation kernel asymptotics for positive-definite Hermitian matrices with poles.

Abstract

We study unitary invariant random matrix ensembles with singular potentials. We obtain asymptotics for the partition functions associated to the Laguerre and Gaussian Unitary Ensembles perturbed with a pole of order $k$ at the origin, in the double scaling limit where the size of the matrices grows, and at the same time the strength of the pole decreases at an appropriate speed. In addition, we obtain double scaling asymptotics of the correlation kernel for a general class of ensembles of positive-definite Hermitian matrices perturbed with a pole. Our results are described in terms of a hierarchy of higher order analogues to the Painlev\'e III equation, which reduces to the Painlev\'e III equation itself when the pole is simple.