Bulk universality for Wigner Hermitian matrices with subexponential decay
László Erdös
Ramírez, José A
Schlein, Benjamin
Tao, Terence
Van, Vu
Yau, Horng-Tzer
In this paper, we consider the ensemble of n×n Wigner Hermitian matrices H = (hℓk)1≤ℓ,k≤n that generalize the Gaussian unitary ensemble (GUE). The matrix elements hℓk = h̄ℓk are given by hℓk = n ?1/2(xℓk + √?1yℓk), where xℓk, yℓk for 1 ≤ ℓ < k ≤ n are i.i.d. random variables with mean zero and variance 1/2, yℓ ℓ = 0 and xℓ ℓ have mean zero and variance 1. We assume the distribution of xℓk, yℓk to have subexponential decay. In [3], four of the authors recently 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 xℓk, yℓk. In [7], the other two authors, using a different method, established the same conclusion assuming instead some moment and support conditions on the xℓk, yℓk. In this short note we observe that the arguments of [3] and [7] 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.
International Press
2010
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doc-type:article
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http://purl.org/coar/resource_type/c_6501
https://research-explorer.ista.ac.at/record/2763
Erdös L, Ramírez J, Schlein B, Tao T, Van V, Yau H. Bulk universality for Wigner Hermitian matrices with subexponential decay. <i>Mathematical Research Letters</i>. 2010;17(4):667-674.
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