Erdos, LaszloSchlein, B.Yau, Horng-Tzer2017-05-182009Erdos, L., B. Schlein, and H.-T. Yau. 2009. “Wegner Estimate and Level Repulsion for Wigner Random Matrices.” International Mathematics Research Notices (August 29). doi:10.1093/imrn/rnp136.1073-7928http://nrs.harvard.edu/urn-3:HUL.InstRepos:32706726We consider N × 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 η ≫ 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.en-USSemicircle lawWigner random matrixlevel repulsionWegner estimatedensity of stateslocalizationextended statesJournal Article2017-05-1810.1093/imrn/rnp136