Rank-uniform local law for Wigner matrices

TL;DR

This paper establishes a comprehensive local law for Wigner matrices that applies to observables of any rank, unifying previous laws, and demonstrates Gaussian fluctuations of quadratic forms on bulk eigenvectors.

Contribution

It introduces a rank-uniform local law for Wigner matrices, extending previous results to arbitrary rank observables and unifying averaged and isotropic local laws.

Findings

Proves a general local law for Wigner matrices of arbitrary rank.

Shows quadratic forms on bulk eigenvectors have approximately Gaussian fluctuations.

Unifies existing local laws into a single comprehensive framework.

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 $A$ 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 $A$ of large rank as well as the result of Benigni and Lopatto [arXiv:2103.12013] valid for specific small rank observables.