Horng-Tzer YauHarvard University
Title: Eigenvalue statistics of random matrices.
We consider N times 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 >> 1/N. This result establishes the semicircle law on the optimal scale. Moreover, we prove that the eigenvalues of a Wigner matrix repel each other, in agreement with the universality conjecture.
