Universality for general Wigner-type matrices

We consider the local eigenvalue distribution of large self-adjoint \(N\times N\) random matrices \(\mathbf{H}=\mathbf{H}^*\) 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, \( \mathrm{diag}(\mathbf{m}(z)) \), where \(\mathbf{m}(z)=(m_1(z),\dots,m_N(z))\) solves the vector equation \( -1/m_i(z) = z + \sum_j s_{ij} m_j(z) \) 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.