Localization near the edge for the lattice Anderson-Bernoulli model on general dimension

Abstract

The Anderson tight-binding model is a fundamental model of quantum transport and localization in disordered media. Completing a problem left open by Bourgain and Kenig, this paper proves Anderson localization near the bottom of the spectrum for the lattice Anderson model with Bernoulli potential, in any dimension $d\ge 2$. The proof uses the multiscale framework of Fr\"ohlich-Spencer and Bourgain-Kenig, and the main new ingredient is a probabilistic discrete unique continuation principle (PDUC) for the discrete Schr\"odinger equation. This PDUC is established via a bootstrap argument and a key probabilistic lemma proved by adaptively revealing the random potential.