Bulk universality for Wigner matrices
Article first published online: 16 FEB 2010
Copyright © 2010 Wiley Periodicals, Inc.
Communications on Pure and Applied Mathematics
Volume 63, Issue 7, pages 895–925, July 2010
How to Cite
Erdős, L., Péché, S., Ramírez, J. A., Schlein, B. and Yau, H.-T. (2010), Bulk universality for Wigner matrices. Comm. Pure Appl. Math., 63: 895–925. doi: 10.1002/cpa.20317
- Issue published online: 14 APR 2010
- Article first published online: 16 FEB 2010
- Manuscript Received: JUN 2009
We consider N × N Hermitian Wigner random matrices H where the probability density for each matrix element is given by the density ν(x) = e−U(x). We prove that the eigenvalue statistics in the bulk are given by the Dyson sine kernel provided that U ∈ C6( \input amssym $\Bbb R$) with at most polynomially growing derivatives and ν(x) ≥ Ce−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. © 2010 Wiley Periodicals, Inc.