Bulk universality for Wigner hermitian matrices with subexponential decay

Abstract

We consider the ensemble of $n\times n$ Wigner hermitian matrices $H=(h_{\ell k}{)}_{1\le \ell ,k\le n}$ 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 $1\le \ell <k\le n$ 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.

9 pages, no figures, submitted, Math. Res. Lett