Universality of random matrices with correlated entries

We consider an N by N real symmetric random matrix X=(xᵢⱼ) where E [xᵢⱼxₖₗ]=ξ ᵢⱼₖₗ. Under the assumption that (ξ ᵢⱼₖₗ) is the discretization of a piecewise Lipschitz function and that the correlation is short-ranged we prove that the empirical spectral measure of X converges to a probability measure. The Stieltjes transform of the limiting measure can be obtained by solving a functional equation. Under the slightly stronger assumption that (xᵢⱼ) has a strictly positive definite covariance matrix, we prove a local law for the empirical measure down to the optimal scale Im z N⁻¹. The local law implies delocalization of eigenvectors. As another consequence we prove that the eigenvalue statistics in the bulk agrees with that of the GOE.