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

Samuel G. G. Johnston

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arXiv · 2601.04869

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

Samuel G. G. Johnston

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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]].

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Samuel G. G. Johnston. 2026-01-08. Log-concavity and concentration bounds for a single gap between GUE eigenvalues. https://arxiv.org/abs/2601.04869

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