Anderson Localization for the hierarchical Anderson-Bernoulli model on $\mathbb{Z}^d$

TL;DR

This paper proves Anderson localization for a hierarchical Bernoulli model on lattices of any dimension, using a method applicable to unique continuation problems.

Contribution

It introduces a new approach to establish Anderson localization in hierarchical Bernoulli models and extends to probabilistic unique continuation on lattices.

Findings

Proves Anderson localization for the hierarchical Anderson-Bernoulli model.

Method is applicable to probabilistic unique continuation on $ abla^d$.

Works for arbitrary lattice dimensions.

Abstract

In this paper, we prove Anderson localization for a hierarchical Anderson-Bernoulli model on lattice with arbitrary dimension, where the potential is characterized by a geometric hierarchical structure combined with fluctuations induced by independent and identically distributed (i.i.d.) Bernoulli random variables. Our method is also applicable to proving a probabilistic unique continuation result on $\mathbb{Z}^d$.