Gaussian Fluctuations of Eigenvalues in Wigner Random Matrices
Open Access Publications from the University of California

## Gaussian Fluctuations of Eigenvalues in Wigner Random Matrices

• Author(s): O'Rourke, Sean
• et al.

## Published Web Location

https://arxiv.org/pdf/0909.2677.pdf
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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.

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