Gaussian Fluctuations of Eigenvalues in Wigner Random Matrices

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.