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International Mathematics Research Notices Advance Access published online on August 29, 2009
International Mathematics Research Notices, doi:10.1093/imrn/rnp136
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© The Author 2009. Published by Oxford University Press. All rights reserved. For Permissions, please e-mail: journals.permissions@oxfordjournals.org

Wegner Estimate and Level Repulsion for Wigner Random Matrices

László Erdos1, Benjamin Schlein2 and Horng-Tzer Yau3

1 Institute of Mathematics, University of Munich, Theresienstr. 39, D-80333 Munich, Germany
2 Department of Pure Mathematics and Mathematical Statistics, University of Cambridge, Wilberforce Rd, Cambridge CB3 0WB, UK
3 Department of Mathematics, Harvard University, Cambridge MA 02138, USA

Correspondence: Correspondence to be sent to: b.schlein{at}dpmms.cam.ac.uk

We consider N x N Hermitian random matrices with independent identically distributed entries (Wigner matrices). The matrices are normalized so that the average spacing between consecutive eigenvalues is of order 1/ N. Under suitable assumptions on the distribution of the single matrix element, we first prove that, away from the spectral edges, the empirical density of eigenvalues concentrates around the Wigner semicircle law on energy scales {eta} >> N–1. This result establishes the semicircle law on the optimal scale and it removes a logarithmic factor from our previous result [6]. We then show a Wegner estimate, i.e., that the averaged density of states is bounded. Finally, we prove that the eigenvalues of a Wigner matrix repel each other, in agreement with the universality conjecture.

Communicated by Prof. Kurt Johansson


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