Eigenstate thermalisation at the edge for Wigner matrices

TL;DR

This paper proves the Eigenstate Thermalisation Hypothesis for Wigner matrices across the entire spectrum, including spectral edges, with optimal fluctuation bounds, extending previous results limited to the bulk or specific observables.

Contribution

It establishes the ETH at spectral edges for Wigner matrices with a new multi-resolvent local law that captures edge scaling effects.

Findings

ETH holds at spectral edges for Wigner matrices.

Introduces a new multi-resolvent local law for edge scaling.

Extends previous bulk-only results to the entire spectrum.

Abstract

We prove the Eigenstate Thermalisation Hypothesis for Wigner matrices uniformly in the entire spectrum, in particular near the spectral edges, with a bound on the fluctuation that is optimal for any observable. This complements earlier works of Cipolloni et. al. (Comm. Math. Phys. 388, 2021; Forum Math., Sigma 10, 2022) and Benigni et. al. (Comm. Math. Phys. 391, 2022; arXiv: 2303.11142) that were restricted either to the bulk of the spectrum or to special observables. As a main ingredient, we prove a new multi-resolvent local law that optimally accounts for the edge scaling.