A "lifting" method for exponential large deviation estimates and an application to certain non-stationary 1D lattice Anderson models

TL;DR

This paper introduces a novel 'lifting' technique to extend exponential large deviation estimates from stationary to certain non-stationary models, enabling new localization results for non-stationary 1D lattice Anderson models.

Contribution

The paper presents a new method to lift large deviation estimates from stationary to non-stationary models, facilitating localization proofs in non-stationary Anderson models.

Findings

Established Anderson localization for certain non-stationary 1D lattice models.

Extended large deviation estimates to non-stationary potentials.

Provided a non-perturbative approach aligned with recent developments.

Abstract

Proofs of localization for random Schr\"odinger operators with sufficiently regular distribution of the potential can take advantage of the fractional moment method introduced by Aizenman-Molchanov, or use the classical Wegner estimate as part of another method, e.g. the multi-scale analysis introduced by Fr\"ohlich-Spencer and significantly developed by Klein and his collaborators. When the potential distribution is singular, most proofs rely crucially on exponential estimates of events corresponding to finite truncations of the operator in question; these estimates in some sense substitute for the classical Wegner estimate. We introduce a method to "lift" such estimates, which have been obtained for many stationary models, to certain closely related non-stationary models. As an application, we use this method to derive Anderson localization on the 1-D lattice for certain non-stationary potentials along the lines of the non-perturbative approach developed by Jitomirskaya-Zhu in 2019.