Bulk universality for Wigner hermitian matrices with subexponential decay

Abstract

In this paper, 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 \cite{ERSY2}, 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 \cite{TVbulk}, 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 \cite{ERSY2} and \cite{TVbulk} 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.