Bulk universality for Wigner hermitian matrices with subexponential decay

TLDR: In this paper, the authors consider the ensemble of Wigner Hermitian matrices and show that the gap distribution and averaged correlation of these matrices are universal under moment and support conditions.

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.