The asymptotic distribution of a single eigenvalue gap of a Wigner matrix

TL;DR

This paper proves that the distribution of a single eigenvalue gap in Wigner matrices converges to the Gaudin-Mehta distribution, extending known results from GUE to more general Wigner ensembles under certain moment conditions.

Contribution

It establishes the asymptotic distribution of a single eigenvalue gap for Wigner matrices, generalizing previous GUE results using the Four Moment Theorem and determinantal process techniques.

Findings

Eigenvalue gap distribution converges to Gaudin-Mehta law in Wigner matrices.

Extension of results from GUE to Wigner ensembles under moment matching.

Established approximate independence of eigenvalue counting and spectrum absence events.

Abstract

We show that the distribution of (a suitable rescaling of) a single eigenvalue gap $\lambda_{i+1}(M_n)-\lambda_i(M_n)$ of a random Wigner matrix ensemble in the bulk is asymptotically given by the Gaudin-Mehta distribution, if the Wigner ensemble obeys a finite moment condition and matches moments with the GUE ensemble to fourth order. This is new even in the GUE case, as prior results establishing the Gaudin-Mehta law required either an averaging in the eigenvalue index parameter $i$, or fixing the energy level $u$ instead of the eigenvalue index. The extension from the GUE case to the Wigner case is a routine application of the Four Moment Theorem. The main difficulty is to establish the approximate independence of the eigenvalue counting function $N_{(-\infty,x)}(\tilde M_n)$ (where $\tilde M_n$ is a suitably rescaled version of $M_n$) with the event that there is no spectrum in an interval $[x,x+s]$, in the case of a GUE matrix. This will be done through some general considerations regarding determinantal processes given by a projection kernel.