Universality for generalized Wigner matrices with Bernoulli distribution

Abstract

The universality for the eigenvalue spacing statistics of generalized Wigner matrices was established in our previous work \cite{EYY} under certain conditions on the probability distributions of the matrix elements. A major class of probability measures excluded in \cite{EYY} are the Bernoulli measures. In this paper, we extend the universality result of \cite{EYY} to include the Bernoulli measures so that the only restrictions on the probability distributions of the matrix elements are the subexponential decay and the normalization condition that the variances in each row sum up to one. The new ingredient is a strong local semicircle law which improves the error estimate on the Stieltjes transform of the empirical measure of the eigenvalues from the order $(N\eta {)}^{-1/2}$ to $(N\eta {)}^{-1}$. Here $\eta$ is the imaginary part of the spectral parameter in the definition of the Stieltjes transform and $N$ is the size of the matrix.

On Sep 17.2011 a small error in the condition of Lemma 6.1 was fixed and accordingly the proof of Thm 6.3 was slighly changed in page 29. (this last change was made after the paper was published, so this version corrects a small error in the published paper)

References in corpus (6)

• Local semicircle law and complete delocalization for Wigner random matrices
• Random matrices: Universality of local eigenvalue statistics up to the edge
• Bulk universality for generalized Wigner matrices
• Universality of Random Matrices and Local Relaxation Flow
• The local relaxation flow approach to universality of the local statistics for random matrices
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• Regularity conditions in the CLT for linear eigenvalue statistics of Wigner matrices
• Universality of Random Matrices and Local Relaxation Flow
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• The Isotropic Semicircle Law and Deformation of Wigner Matrices
• Average Density of States for Hermitian Wigner Matrices
• Spectral Properties of Wigner Matrices