Gap Probabilities for Double Intervals in Hermitian Random Matrix Ensembles as $\tau$-Functions -- Spectrum Singularity case

TL;DR

This paper studies the probability of eigenvalue exclusion in Hermitian random matrices with a Bessel kernel, showing it equals the square of a Painlevé ext{PIII} au-function and factorizes into two ext{PIIIdash} au-functions, revealing deep connections between these systems.

Contribution

It establishes a novel link between eigenvalue exclusion probabilities and Painlevé ext{PIII} and ext{PIIIdash} au-functions, extending previous results with new factorization insights.

Findings

Probability expressed as square of ext{PIII} au-function

Factorization into two ext{PIIIdash} au-functions

Identities between products of ext{PIIIdash} au-functions

Abstract

The probability for the exclusion of eigenvalues from an interval $(-x,x)$ symmetrical about the origin for a scaled ensemble of Hermitian random matrices, where the Fredholm kernel is a type of Bessel kernel with parameter $ a $ (a generalisation of the sine kernel in the bulk scaling case), is considered. It is shown that this probability is the square of a $\tau$-function, in the sense of Okamoto, for the Painlev\'e system \PIII. This then leads to a factorisation of the probability as the product of two $\tau$-functions for the Painlev\'e system \PIIIdash. A previous study has given a formula of this type but involving \PIIIdash systems with different parameters consequently implying an identity between products of $\tau$-functions or equivalently sums of Hamiltonians.