Gaussian Fluctuations of Eigenvalues in Wigner Random Matrices

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 x k denote eigenvalue number k. Under the condition that both k and n−k tend to infinity as n→∞, we show that x k is normally distributed in the limit. We also consider the joint limit distribution of eigenvalues (x_k₁,…,x_kₘ) from the GOE or GSE where k 1, n−k m and k i+1−k i , 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.