On localization for the alloy-type Anderson-Bernoulli model with long-range hopping

TL;DR

This paper proves Anderson localization near the spectral edge for a class of alloy-type Bernoulli models with long-range hopping on bZ^d, extending prior methods and applicable to analogous models on bR^d.

Contribution

It extends multi-scale analysis techniques to alloy-type Bernoulli models with long-range hopping, combining Bourgain's and Klopp's methods for initial scale estimates.

Findings

Proves Anderson localization near spectral edges for the model.

Extends Bourgain's multi-scale analysis to new models.

Applicable to analogous models on bR^d.

Abstract

In this paper, we prove the Anderson localization near the spectral edge for some alloy-type Anderson-Bernoulli model on $\mathbb{Z}^d$ with exponential long-range hopping. This extends the work of Bourgain [Geometric Aspects of Functional Analysis, LNM 1850: 77--99, 2004], in which he pioneered a novel multi-scale analysis to treat Bernoulli random variables. Our proof is mainly based on Bourgain's method. However, to establish the initial scales Green's function estimates, we adapt the approach of Klopp [Comm. Math. Phys, Vol. 232, 125--155, 2002], which is based on the Floquet-Bloch theory and a certain quantitative uncertainty principle. Our proof also applies to an analogues model on $\mathbb{R}^d.$