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

2026/01/08 by Samuel G. G. Johnston · 1 voice
Mathematics · #math.PR

paper · pdf

arxiv published 2026/01/08 · arxiv updated 2026/01/09

Abstract

We observe that the distribution of the eigenvalues of an N-by-N GUE random matrix is log-concave on 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]].

Citations

• On the distribution of eigenvalues of GUE and its minors at fixed index
• On the limit of random hives with GUE boundary conditions
• The macroscopic shape of Gelfand-Tsetlin patterns and free probability
• Sums of GUE matrices and concentration of hives from correlation decay of eigengaps
• Quantitative limit theorems via relative log-concavity
• Concentration inequalities for log-concave sequences
• Large deviations for random hives and the spectrum of the sum of two random matrices
• The asymptotic distribution of a single eigenvalue gap of a Wigner matrix
• Sur la loi limite de l'espacement des valeurs propres d'une matrice ale´atoire

Discussions

• arxiv.org/abs/2601.04869 'Log-concavity and concentration bounds for a single gap between GUE eigenvalues' - Samuel G. G. Johnston [bsky, 2 points, 0 comments]