# Gaussian Fluctuations of Eigenvalues in Wigner Random Matrices - Mathematics > Probability

Abstract: We study the fluctuations of eigenvalues from a class of Wigner randommatrices that generalize the Gaussian orthogonal ensemble. We begin byconsidering an $n \times n$ matrix from the Gaussian orthogonal ensemble GOEor 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 $n$, weshow that $x k$ is normally distributed in the limit. We also consider thejoint limit distribution of $m$ eigenvalues from the GOE or GSE with similarconditions on the indices. The result is an $m$-dimensional normaldistribution. Using a recent universality result by Tao and Vu, we extend ourresults to a class of Wigner real symmetric matrices with non-Gaussian entriesthat have an exponentially decaying distribution and whose first four momentsmatch the Gaussian moments.

Author: Sean O'Rourke

Source: https://arxiv.org/