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

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.