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

TL;DR

The paper shows that the joint law of GUE eigenvalues is log-concave, which implies log-concavity for the single bulk gap after semicircle renormalization. By combining log-concavity with convex-order comparisons and Tao’s existing bounds, it derives sharper concentration results for the semicircle-renormalised eigengaps $g_i^{(N)}$, including exponential tails, unit-mean moments, and strong tail controls for sums of consecutive gaps. It further argues that the Gaudin–Mehta distribution and the Gelfand–Tsetlin patterns inherit log-concavity properties, reinforcing the broader applicability of the approach to eigenvalue gaps and related combinatorial structures. The results provide stronger, uniform-tailed estimates in the bulk and establish a pathway to Gaussian-type tails via relative log-concavity and limit processes.

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].

Theorems & Definitions (16)

• Theorem 1.1: Theorem 1.1 of tao
• Theorem 1.2
• Theorem 1.3
• Remark 1
• Proposition 2.1
• Remark 2
• Proposition 2.2: Variant of results in MMLV
• proof : Proof of Theorem \ref{['thm:main2']}
• Theorem 2.3
• proof