Gap probability at the hard edge for random matrix ensembles with pole singularities in the potential

TL;DR

This paper analyzes the asymptotic behavior of the eigenvalue gap probability at the hard edge of certain random matrix ensembles with pole singularities, using Riemann-Hilbert techniques and Painlevé equations.

Contribution

It introduces a new Fredholm determinant related to higher order Painlevé III equations for ensembles with pole perturbations, providing asymptotics and explicit formulas.

Findings

Derived large s asymptotics of the Fredholm determinant.

Expressed the determinant in terms of a coupled Painlevé III system.

Connected the gap probability to integrable systems and Riemann-Hilbert analysis.

Abstract

We study the Fredholm determinant of an integrable operator acting on the interval $(0,s)$ whose kernel is constructed out of a hierarchy of higher order analogues to the Painlev\'{e} III equation. This Fredholm determinant describes the critical behavior of the eigenvalue gap probability at the hard edge of unitary invariant random matrix ensembles perturbed by poles of order $k$ in the double scaling regime. Using the Riemann-Hilbert method, we obtain the large $s$ asymptotics of the Fredholm determinant. Moreover, we derive a Painlev\'e type formula of the Fredholm determinant, which is expressed in terms of an explicit integral involving a solution to the coupled Painlev\'e III system.