Mathematics > Probability
[Submitted on 15 Sep 2009 (v1), last revised 10 Mar 2010 (this version, v3)]
Title:Gaussian Fluctuations of Eigenvalues in Wigner Random Matrices
View PDFAbstract: 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.
Submission history
From: Sean O'Rourke [view email][v1] Tue, 15 Sep 2009 00:18:00 UTC (16 KB)
[v2] Thu, 17 Sep 2009 14:28:17 UTC (16 KB)
[v3] Wed, 10 Mar 2010 20:33:38 UTC (16 KB)
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