Large deviations of the Lyapunov exponent and localization for the 1D Anderson model

TL;DR

This paper presents a simplified proof of Anderson localization for the 1D Anderson model using large deviation estimates and subharmonicity, and introduces a general uniform bound relevant for dynamical localization.

Contribution

It offers a concise proof of localization in the 1D Anderson model leveraging large deviation techniques and develops a broad, uniform Craig-Simon bound for dynamical localization.

Findings

Proof of Anderson localization using large deviations

Development of a uniform Craig-Simon bound

Applicable to high generality cases

Abstract

The proof of Anderson localization for the 1D Anderson model with arbitrary (e.g. Bernoulli) disorder, originally given by Carmona-Klein-Martinelli in 1987, is based in part on the multi-scale analysis. Later, in the 90s, it was realized that for one-dimensional models with positive Lyapunov exponents some parts of multi-scale analysis can be replaced by considerations involving subharmonicity and large deviation estimates for the corresponding cocycle, leading to nonperturbative proofs for 1D quasiperiodic models. In this paper we present a short proof along these lines, for the Anderson model. To prove dynamical localization we also develop a uniform version of Craig-Simon's bound that works in high generality and may be of independent interest.