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Smallest Gaps between Eigenvalues of Wigner Matrices without Smoothness Assumptions

Primarily AI-generated textHuman understanding: some partsmath.PR — Probability

Contributed by Patrick Lopatto ↗

Version 1 / Oct 31, 2026 / CC BY 4.0

Abstract

We determine the asymptotic distribution of the smallest gaps between eigenvalues in the bulk of the spectrum of real symmetric Wigner matrices, including Bernoulli matrices. After multiplication by N3/2N^{3/2}, the gaps and their locations converge to a Poisson point process with the same intensity as for the Gaussian orthogonal ensemble. Since changing a single entry can move eigenvalues by an amount comparable to the smallest gaps, standard perturbative comparison estimates do not directly resolve this scale for arbitrary entry distributions. Instead, we use that resampling a few random entries does not change the law of the matrix, and we show that the resampling moves each pair of nearby eigenvalues by a random walk that satisfies a local limit theorem; this requires excluding arithmetic structure in the eigenvectors. We also prove a law of large numbers for the number of gaps at all scales between N−3/2N^{-3/2} and N−1N^{-1}, and we sketch the proof of the analogous smallest-gap result for complex Hermitian Wigner matrices.

Provenance statement

Proof discovery, verification, and writing were done with GPT 6 Astra, GPT 6.1 Sol, and Claude Opus 5.5.

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Version history

  1. v1Submitted by Patrick LopattoInitial depositCurrentOct 30, 2026