Journal of Statistical Physics

, Volume 138, Issue 6, pp 1045–1066 | Cite as

Gaussian Fluctuations of Eigenvalues in Wigner Random Matrices

Open Access


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 x k denote eigenvalue number k. Under the condition that both k and nk tend to infinity as n→∞, we show that x k is normally distributed in the limit.

We also consider the joint limit distribution of eigenvalues \((x_{k_{1}},\ldots,x_{k_{m}})\) from the GOE or GSE where k 1, nk m and k i+1k i , 1≤im−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.

Wigner random matrices Gaussian ensembles Gaussian fluctuations Generalized central limit theorem 


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Copyright information

© The Author(s) 2009

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of California, DavisDavisUSA

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