# Bulk universality for Wigner Hermitian matrices with subexponential decay

Erdös L, Ramírez J, Schlein B, Tao T, Van V, Yau H. 2010. Bulk universality for Wigner Hermitian matrices with subexponential decay. Mathematical Research Letters. 17(4), 667–674.

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Erdös, László

^{ISTA}^{}; Ramírez, José A; Schlein, Benjamin; Tao, Terence; Van, Vu; Yau, Horng-TzerAbstract

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.

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Date Published

2010-07-01

Journal Title

Mathematical Research Letters

Volume

17

Issue

4

Page

667 - 674

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### Cite this

Erdös L, Ramírez J, Schlein B, Tao T, Van V, Yau H. Bulk universality for Wigner Hermitian matrices with subexponential decay.

*Mathematical Research Letters*. 2010;17(4):667-674.Erdös, L., Ramírez, J., Schlein, B., Tao, T., Van, V., & Yau, H. (2010). Bulk universality for Wigner Hermitian matrices with subexponential decay.

*Mathematical Research Letters*. International Press.Erdös, László, José Ramírez, Benjamin Schlein, Terence Tao, Vu Van, and Horng Yau. “Bulk Universality for Wigner Hermitian Matrices with Subexponential Decay.”

*Mathematical Research Letters*. International Press, 2010.L. Erdös, J. Ramírez, B. Schlein, T. Tao, V. Van, and H. Yau, “Bulk universality for Wigner Hermitian matrices with subexponential decay,”

*Mathematical Research Letters*, vol. 17, no. 4. International Press, pp. 667–674, 2010.Erdös, László, et al. “Bulk Universality for Wigner Hermitian Matrices with Subexponential Decay.”

*Mathematical Research Letters*, vol. 17, no. 4, International Press, 2010, pp. 667–74.