Ergodicity and dynamical localization for Delone-Anderson operators

TL;DR

This paper investigates the ergodic properties and spectral behavior of Delone-Anderson operators, establishing the existence of the integrated density of states and demonstrating dynamical localization through Lifshitz-tail estimates and multi-scale analysis.

Contribution

It introduces new methods for analyzing ergodicity, spectrum, and localization in Delone-Anderson operators using geometric and probabilistic techniques.

Findings

Existence of integrated density of states for Delone-Anderson operators.

Conditions under which the spectrum is almost surely determined.

Proof of dynamical localization at the bottom of the spectrum.

Abstract

We study the ergodic properties of Delone-Anderson operators, using the framework of randomly coloured Delone sets and Delone dynamical systems. In particular, we show the existence of the integrated density of states and, under some assumptions on the geometric complexity of the underlying Delone sets, we obtain information on the almost-sure spectrum of the family of random operators. We then exploit these results to study the Lifshitz-tail behaviour of the integrated density of states of a Delone-Anderson operator at the bottom of the spectrum. Furthermore, we use Lifshitz-tail estimates as an input for the multi-scale analysis to prove dynamical localization.