Mathematical Research Letters

Volume 17 (2010)

Number 4

Bulk universality for Wigner hermitian matrices with subexponential decay

Pages: 667 – 674



László Erdös (University of Munich, Germany)

José Ramírez (Universidad de Costa Rica, San Jose, Costa Rica)

Benjamin Schlein (University of Cambridge, Cambridge, UK)

Terence Tao (University of California at Los Angeles, Los Angeles CA)

van Vu (Rutgers, NJ)

Horng-Tzer Yau (Harvard University, Cambridge MA)


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.

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