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×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 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, n−k m and k i+1−k i , 1≤i≤m−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.
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O’Rourke, S. Gaussian Fluctuations of Eigenvalues in Wigner Random Matrices. J Stat Phys 138, 1045–1066 (2010). https://doi.org/10.1007/s10955-009-9906-y
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DOI: https://doi.org/10.1007/s10955-009-9906-y