Largest gaps between bulk eigenvalues of unitary-invariant random Hermitian matrices

TL;DR

This paper analyzes the largest gaps between eigenvalues in the bulk of unitary-invariant Hermitian matrices, showing their rescaled distribution converges to a gamma-Gumbel distribution, extending previous Gaussian results.

Contribution

It extends the understanding of eigenvalue gaps to more general potentials beyond the Gaussian case, providing explicit distributional limits for the largest gaps.

Findings

Rescaled eigenvalue gaps converge to a gamma-Gumbel distribution.

Explicit dependence of the limit distribution on the potential and interval.

Generalization of previous Gaussian potential results to broader classes of ensembles.

Abstract

We study $n\times n$ random Hermitian matrix ensembles that are invariant under unitary conjugation. Let $I$ be a finite union of intervals lying in the bulk, and let $m_{k}^{(n)}$ be the $k$-th largest gap between consecutive eigenvalues lying in $I$. We prove that the rescaled gap $\smash{\tau_{k}^{(n)}}$, which is defined by \begin{align*} m_{k}^{(n)} = \frac{1}{2\pi \inf_{I}\rho} \bigg( \frac{\sqrt{32 \log n}}{n} + \frac{3q-8}{2q} \frac{ \log(2\log n)}{n \sqrt{2\log n}} + \frac{4\tau_{k}^{(n)}}{n \sqrt{2\log n}} \bigg), \end{align*} converges in distribution as $n\to +\infty$ to a gamma-Gumbel random variable that is shifted by an explicit constant $c_{V,I}$ depending only on $I$ and on the potential $V$. Here $\rho$ is the density of the equilibrium measure and $q\in \mathbb{N}_{>0}$ is the highest order at which $\rho(x)$ approaches $\inf_{I}\rho$ with $x\in I$; for example, if $\rho(x)=1/(\pi\sqrt{x(1-x)})$, then $q=2$ if $\frac{1}{2}\in \overline{I}$ and $q=1$ otherwise. This work extends a result of Feng and Wei beyond the Gaussian potential.