Rank-uniform local law for Wigner matrices
Abstract
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 on the bulk eigenvectors of a Wigner matrix has approximately Gaussian fluctuation. For the bulk spectrum, we thus generalize our previous result [arXiv:2103.06730] valid for test matrices of large rank as well as the result of Benigni and Lopatto [arXiv:2103.12013] valid for specific small rank observables.
Cite
@article{arxiv.2203.01861,
title = {Rank-uniform local law for Wigner matrices},
author = {Giorgio Cipolloni and László Erdős and Dominik Schröder},
journal= {arXiv preprint arXiv:2203.01861},
year = {2023}
}
Comments
We corrected the a priori estimate (Eqs (3.76)-(3.84)) in the proof of Theorem 2.2. which previously was an unnecessary overestimate. As a consequence the argument for the final bound in (3.85)-(3.86) could be simplified