Universality for general Wigner-type matrices

We consider the local eigenvalue distribution of large self-adjoint $N\times N$randommatrices$$\mathbf{H}$=$\mathbf{H}$^*$ with centered independent entries. In contrast to previous works the matrix of variances $s_{ij} = $\mathbb{E}$, |h_{ij}|^2 $ is not assumed to be stochastic. Hence the density of states is not the Wigner semicircle law. Its possible shapes are described in the companion paper [1]. We show that as $N$ grows, the resolvent, $\mathbf{G}(z)=(\mathbf{H}-z{)}^{-1}$, converges to a diagonal matrix, $\mathrm{d}\mathrm{i}\mathrm{a}\mathrm{g}(\mathbf{m}(z))$, where $\mathbf{m}(z)=(m_1(z),\dots ,m_N(z))$ solves the vector equation $-1\mathrm{/}{m}_i(z)=z+{\sum }_j{s}_{ij}{m}_j(z)$ that has been analyzed in [1]. We prove a local law down to the smallest spectral resolution scale, and bulk universality for both real symmetric and complex hermitian symmetry classes.