Bulk universality for generalized Wigner matrices

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 ${cle N sigma_{ij}^2 le c^{-1}}$ 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.