  We consider an $N$ by $N$ real symmetric random matrix $X=(x_{ij})$ where
$\mathbb{E}x_{ij}x_{kl}=\xi_{ijkl}$. Under the assumption that $(\xi_{ijkl})$
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 $(x_{ij})$ has a strictly positive definite covariance
matrix, we prove a local law for the empirical measure down to the optimal
scale $\text{Im} z \gtrsim N^{-1}$. 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.
