Optimal multi-resolvent local laws for Wigner matrices

Summary

We prove local laws for arbitrary products of resolvents of a Wigner random matrix with deterministic matrices in between. Our key finding is that the size of such products heavily depends on whether some of the deterministic matrices are traceless.

Abstract

We prove local laws, i.e. optimal concentration estimates for arbitrary products of resolvents of a Wigner random matrix with deterministic matrices in between. We find that the size of such products heavily depends on whether some of the deterministic matrices are traceless. Our estimates correctly account for this dependence and they hold optimally down to the smallest possible spectral scale.

Paper

2112.13693.pdf