From gap probabilities in random matrix theory to eigenvalue expansions

TL;DR

This paper develops a method to analyze the asymptotic behavior of eigenvalues of certain integral operators in random matrix theory, linking spectral properties to Fredholm determinants, with explicit results for Airy and Bessel kernels.

Contribution

It introduces a new approach to derive eigenvalue asymptotics for integrable operators, connecting spectral analysis with Fredholm determinants in a novel way.

Findings

Asymptotic formulas for eigenvalues of Airy kernel

Asymptotic formulas for eigenvalues of Bessel kernel

Link between eigenvalue behavior and Fredholm determinants

Abstract

We present a method to derive asymptotics of eigenvalues for trace-class integral operators $K:L^2(J;d\lambda)\circlearrowleft$, acting on a single interval $J\subset\mathbb{R}$, which belong to the ring of integrable operators \cite{IIKS}. Our emphasis lies on the behavior of the spectrum $\{\lambda_i(J)\}_{i=0}^{\infty}$ of $K$ as $|J|\rightarrow\infty$ and $i$ is fixed. We show that this behavior is intimately linked to the analysis of the Fredholm determinant $\det(I-\gamma K)|_{L^2(J)}$ as $|J|\rightarrow\infty$ and $\gamma\uparrow 1$ in a Stokes type scaling regime. Concrete asymptotic formul\ae\, are obtained for the eigenvalues of Airy and Bessel kernels in random matrix theory.