Bulk Universality for Wigner Matrices

Abstract

We consider $N\times N$ Hermitian Wigner random matrices $H$ where the probability density for each matrix element is given by the density $\nu (x)=e^{-U(x)}$. We prove that the eigenvalue statistics in the bulk is given by Dyson sine kernel provided that $U \in C^6(\RR)$ with at most polynomially growing derivatives and $\nu (x)\le C{e}^{-C|x|}$ for $x$ large. The proof is based upon an approximate time reversal of the Dyson Brownian motion combined with the convergence of the eigenvalue density to the Wigner semicircle law on short scales.

23 pages, 1 figure. An error in the previous version has been corrected

References in corpus (3)

• Local semicircle law and complete delocalization for Wigner random matrices
• Universality of the local spacing distribution in certain ensembles of hermitian Wigner matrices
• Random matrices: Universality of local eigenvalue statistics

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