# Bulk universality for Wigner hermitian matrices with subexponential
decay

Research paper by **Laszlo Erdos, Jose Ramirez, Benjamin Schlein, Terence Tao, Van Vu, Horng-Tzer Yau**

Indexed on: **30 Jun '10**Published on: **30 Jun '10**Published in: **Mathematics - Probability**

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#### 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.