Mathematics

The resolvent of a large dimensional self-adjoint random matrix with correlated entries on short scales converges weakly to a non-random matrix that satisfies the Matrix Dyson Equation. We present a comprehensive stability analysis of this non-linear matrix equation in asymptotically infinite dimensions down to the length scale of the eigenvalue spacing. This analysis is then used to show that the local eigenvalue statistics are universal, i.e., they do not depend on the distribution of the entries of the random matrix under consideration (Wigner-Dyson-Mehta spectral universality).