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GAUSSIAN FLUCTUATIONS OF EIGENVALUES IN WIGNER RANDOM MATRICES

By Sean O&apos

Abstract

We study the fluctuations of eigenvalues from a class of Wigner random matrices that generalize the Gaussian orthogonal ensemble. We begin by considering an n × n matrix from the Gaussian orthogonal ensemble (GOE) or Gaussian symplectic ensemble (GSE) and let xk denote eigenvalue number k. Under the condition that both k and n − k tend to infinity as n → ∞, we show that xk is normally distributed in the limit. We also consider the joint limit distribution of eigenvalues (xk1,..., xkm from the GOE or GSE where k1, n − km and ki+1 − ki, 1 ≤ i ≤ m − 1, tend to infinity with n. The result in each case is an m-dimensional normal distribution. Using a recent universality result by Tao and Vu, we extend our results to a class of Wigner real symmetric matrices with non-Gaussian entries that have an exponentially decaying distribution and whose first four moments match the Gaussian moments

Topics: 1.1. Real Symmetric Wigner Matrices. Following Tao and Vu in [21, we define
Year: 2009
OAI identifier: oai:CiteSeerX.psu:10.1.1.246.3550
Provided by: CiteSeerX
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