Universality for general Wigner-type matrices

Abstract

We consider the local eigenvalue distribution of large self-adjoint Ntimes N random matrices mathbf {H}=mathbf {H}^* with centered independent entries. In contrast to previous works the matrix of variances s_{i j}=mathbbm {E} |h_{i j}|^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 (Ajanki et al. in Quadratic Vector Equations on the Complex Upper Half Plane. arXiv:1506.05095). 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_{i j} m_j(z) that has been analyzed in Ajanki et al. (Quadratic Vector Equations on the Complex Upper Half Plane. arXiv:1506.05095). We prove a local law down to the smallest spectral resolution scale, and bulk universality for both real symmetric and complex hermitian symmetry classes.