Bulk universality for Wigner hermitian matrices with subexponential decay

TL;DR

This paper proves that the local spectral statistics of Wigner hermitian matrices with subexponential decay are universal, matching those of the Gaussian Unitary Ensemble, without additional regularity or moment assumptions.

Contribution

It combines previous methods to establish universality for all Wigner matrices with subexponential decay, removing extra assumptions required in earlier works.

Findings

Universality of gap distribution confirmed for all such matrices

Averaged k-point correlations match GUE results

No additional regularity assumptions needed

Abstract

We consider the ensemble of $n \times n$ Wigner hermitian matrices $H = (h_{\ell k})_{1 \leq \ell,k \leq n}$ that generalize the Gaussian unitary ensemble (GUE). The matrix elements $h_{k\ell} = \bar h_{\ell k}$ are given by $h_{\ell k} = n^{-1/2} (x_{\ell k} + \sqrt{-1} y_{\ell k})$, where $x_{\ell k}, y_{\ell k}$ for $1 \leq \ell < k \leq n$ are i.i.d. random variables with mean zero and variance 1/2, $y_{\ell\ell}=0$ and $x_{\ell \ell}$ have mean zero and variance 1. We assume the distribution of $x_{\ell k}, y_{\ell k}$ to have subexponential decay. In a recent paper, four of the authors recently established that the gap distribution and averaged $k$-point correlation of these matrices were \emph{universal} (and in particular, agreed with those for GUE) assuming additional regularity hypotheses on the $x_{\ell k}, y_{\ell k}$. In another recent paper, the other two authors, using a different method, established the same conclusion assuming instead some moment and support conditions on the $x_{\ell k}, y_{\ell k}$. In this short note we observe that the arguments of these two papers can be combined to establish universality of the gap distribution and averaged $k$-point correlations for all Wigner matrices (with subexponentially decaying entries), with no extra assumptions.