Localization for the random displacement model

TL;DR

This paper establishes spectral and dynamical localization for a multi-dimensional random displacement model near the spectrum's bottom, extending Lifshitz tail bounds and proving a new Wegner estimate using multiscale analysis.

Contribution

It introduces a novel approach to prove localization in the random displacement model by extending Lifshitz tail bounds and developing a new Wegner estimate, utilizing a property of related Neumann problems.

Findings

Spectral localization near the spectrum's bottom.

Dynamical localization established for the model.

Extension of Lifshitz tail bounds to this setting.

Abstract

We prove spectral and dynamical localization for the multi-dimensional random displacement model near the bottom of its spectrum by showing that the approach through multiscale analysis is applicable. In particular, we show that a previously known Lifshitz tail bound can be extended to our setting and prove a new Wegner estimate. A key tool is given by a quantitative form of a property of a related single-site Neumann problem which can be described as "bubbles tend to the corners".