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 νᵢⱼ with a subexponential decay. Let σᵢⱼ² be the variance for the probability measure νᵢⱼ with the normalization property that ∑ᵢ σ²ᵢⱼ = 1 for all j. Under essentially the only condition that c≤ N σᵢⱼ² ≤ 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.