Rank-uniform local law for Wigner matrices

Giorgio Cipolloni, László Erdős, Dominik Schröder

Published 2022-03-03Version 1

We prove a general local law for Wigner matrices which optimally handles observables of arbitrary rank and thus it unifies the well-known averaged and isotropic local laws. As an application, we prove that the quadratic forms of a general deterministic matrix $A$ on the bulk eigenvectors of a Wigner matrix has approximately Gaussian fluctuation. For the bulk spectrum, we thus generalize our previous result \cite{2103.06730} valid for test matrices $A$ of large rank as well as the result of Benigni and Lopatto \cite{2103.12013} valid for specific small rank observables.