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

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 $\mathbb{Z}^d$.