Erdős, L. et al. (2009) “Wegner Estimate and Level Repulsion for Wigner Random Matrices,” Wegner Estimate and Level Repulsion for Wigner Random Matrices

We consider N × N Hermitian randommatrices with independent identically distributed entries (Wigner matrices). The matrices are normalized so that the average spacing be-tween consecutive eigenvalues is of order 1/N. Under suitable assumptions on the dis-tribution 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 Weg-ner 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.