The Anderson model with missing sites

TL;DR

This paper proves dynamical localization at the bottom of the spectrum for an Anderson model with missing sites on a Delone set in any dimension, using a spatial averaging approach and multiscale analysis.

Contribution

It extends localization results to non-ergodic Anderson models with missing sites defined on Delone sets without geometric complexity assumptions.

Findings

Established dynamical localization at the lower spectral edge.

Derived a uniform Wegner estimate depending on Delone set parameters.

Applied multiscale analysis to non-ergodic models for localization proof.

Abstract

In the present note we show dynamical localization for an Anderson model with missing sites in a discrete setting at the bottom of the spectrum in arbitrary dimension $d$. In this model, the random potential is defined on a relatively dense subset of $\mathbb Z^d$, not necessarily periodic, i.e., a Delone set in $\mathbb Z^d$. To work in the lower band edge we need no further assumption on the geometric complexity of the Delone set. We use a spatial averaging argument by Bourgain-Kenig to obtain a uniform Wegner estimate and an initial length scale estimate, which yields localization through the Multiscale Analysis for non ergodic models. This argument gives an explicit dependence on the maximal distance parameter of the Delone set for the Wegner estimate. We discuss the case of the upper spectral band edge and the arising need of imposing the (complexity) condition of strict uniform pattern frequency on the Delone set.