Gaussian Fluctuations of Eigenvalues in Wigner Random Matrices

TL;DR

This paper investigates the asymptotic normality of eigenvalue fluctuations in Wigner random matrices, extending classical Gaussian results to broader classes with non-Gaussian entries under certain moment conditions.

Contribution

It proves the normal distribution of eigenvalues in Wigner matrices and extends these results to non-Gaussian matrices using universality principles.

Findings

Eigenvalues are normally distributed when both index and matrix size tend to infinity.

Joint distribution of multiple eigenvalues converges to a multivariate normal.

Results apply to non-Gaussian Wigner matrices with matching moments and exponential decay.

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 $n \times n$ 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 with $n$, we show that $x_k$ is normally distributed in the limit. We also consider the joint limit distribution of $m$ eigenvalues from the GOE or GSE with similar conditions on the indices. The result 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.