Eigenstate thermalisation at the edge for Wigner matrices

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.