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

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]].

8 pages