Large gap asymptotics at the hard edge for product random matrices and Muttalib-Borodin ensembles

TL;DR

This paper investigates the asymptotic behavior of the smallest eigenvalue distribution at the hard edge for certain large random matrices, using advanced integral operators and Riemann-Hilbert problem techniques.

Contribution

It introduces a novel approach to analyze large gap asymptotics for product random matrices and Muttalib-Borodin ensembles via Fredholm determinants and Riemann-Hilbert problems.

Findings

Derived large gap asymptotics for eigenvalue distributions

Expressed logarithmic derivatives in terms of Riemann-Hilbert problems

Generalized the hard edge Bessel kernel results

Abstract

We study the distribution of the smallest eigenvalue for certain classes of positive-definite Hermitian random matrices, in the limit where the size of the matrices becomes large. Their limit distributions can be expressed as Fredholm determinants of integral operators associated to kernels built out of Meijer $G$-functions or Wright's generalized Bessel functions. They generalize in a natural way the hard edge Bessel kernel Fredholm determinant. We express the logarithmic derivatives of the Fredholm determinants identically in terms of a $2\times 2$ Riemann-Hilbert problem, and use this representation to obtain the so-called large gap asymptotics.