Log-concavity and concentration bounds for a single gap between GUE eigenvalues

TL;DR

This paper proves that the distribution of GUE eigenvalues and their gaps is log-concave, leading to improved concentration bounds for eigenvalue gaps compared to previous results.

Contribution

It establishes log-concavity of eigenvalue distributions and single gap distributions in GUE, enabling stronger concentration bounds than prior work.

Findings

Eigenvalue distribution of GUE is log-concave.

Single eigenvalue gap distribution is log-concave.

Derived improved concentration bounds for eigengaps.

Abstract

We observe that the distribution of the eigenvalues of an $N$-by-$N$ GUE random matrix is log-concave on $\mathbb{R}^N$, and that the same is true for the law of a single gap between two consecutive eigenvalues. We use this observation to prove several concentration bounds for the semicircle-renormalised eigengaps, improving on bounds recently obtained in [Tao (2024). On the distribution of eigenvalues of GUE and its minors at fixed index. [arXiv:2412.10889]].