Large gaps between random eigenvalues

TL;DR

This paper analyzes the distribution of eigenvalues in large random matrices, providing a corrected formula for the probability of eigenvalue gaps and introducing new stochastic calculus techniques.

Contribution

It offers a corrected asymptotic formula for eigenvalue gap probabilities in beta-ensembles and employs a novel Brownian carousel representation with Girsanov transformation.

Findings

Derived the asymptotic probability of no eigenvalues in a fixed interval.

Introduced the Brownian carousel representation for the limit process.

Applied stochastic calculus methods to prove the results.

Abstract

We show that in the point process limit of the bulk eigenvalues of $\beta$-ensembles of random matrices, the probability of having no eigenvalue in a fixed interval of size $\lambda$ is given by \[\bigl(\ kappa_{\beta}+o(1)\bigr)\lambda^{\gamma_{\beta}}\exp\biggl(-{\bet a}{64}\lambda^2+\biggl({\beta}{8}-{1}{4}\biggr)\lambda\biggr)\] as $\lambda\to\infty$, where \[\gamma_{\beta}={1}{4}\biggl({\beta}{2}+{2}{\beta}-3\biggr)\] and $\kappa_{\beta}$ is an undetermined positive constant. This is a slightly corrected version of a prediction by Dyson [J. Math. Phys. 3 (1962) 157--165]. Our proof uses the new Brownian carousel representation of the limit process, as well as the Cameron--Martin--Girsanov transformation in stochastic calculus.