Bulk universality for generalized Wigner matrices

Abstract

Consider N × N Hermitian or symmetric random matrices H where the distribution of the (i, j) matrix element is given by a probability measure ν ij with a subexponential decay. Let \({\sigma_{ij}^2}\) be the variance for the probability measure ν ij with the normalization property that \({\sum_{i} \sigma^2_{ij} = 1}\) for all j. Under essentially the only condition that \({c\le N \sigma_{ij}^2 \le c^{-1}}\) for some constant c > 0, we prove that, in the limit N → ∞, the eigenvalue spacing statistics of H in the bulk of the spectrum coincide with those of the Gaussian unitary or orthogonal ensemble (GUE or GOE). We also show that for band matrices with bandwidth M the local semicircle law holds to the energy scale M −1.