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Bulk universality for Wigner Hermite matrices with subexponential decay. (English) Zbl 1277.15027

Summary: 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 \(h_{\ell k}=\bar{h}_{\ell k}\) 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 [Erdős, Ramírez, Schlein, Yau, and S. Péché, Commun. Pure Appl. Math. 63, No. 7, 895–925 (2010; Zbl 1216.15025)] established that the gap distribution and averaged \(k\)-point correlation of these matrices were 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 [Tao and Vu, Commun. Math. Phys. 298, No. 2, 549–572 (2010; Zbl 1202.15038)], 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.

MSC:

15B52 Random matrices (algebraic aspects)
60B20 Random matrices (probabilistic aspects)
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