A Note on the Central Limit Theorem for the Eigenvalue Counting Function of Wigner Matrices

Details A Note on the Central Limit Theorem for the Eigenvalue Counting Function of Wigner Matrices A Note on the Central Limit Theorem for the Eigenvalue Counting Function of Wigner Matrices

2011-01-13

arxiv.org/abs/1101.2553v1

Mathematics

Probability

Scientific paper

The purpose of this note is to establish a Central Limit Theorem for the number of eigenvalues of a Wigner matrix in an interval. The proof relies on the correct aymptotics of the variance of the eigenvalue counting function of GUE matrices due to Gustavsson, and its extension to large families of Wigner matrices by means of the Tao and Vu Four Moment Theorem and recent localization results by Erd\"os, Yau and Yin.

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