Dynamical Localization for the Singular Anderson Model in $\mathbb{Z}^d$

TL;DR

This paper demonstrates that with a single-energy multiscale analysis, one can derive strong dynamical localization results for the Anderson model on $ olinebreak bZ^d$, including recent cases with Bernoulli distributions.

Contribution

It shows that a single-energy multiscale analysis suffices to establish strong dynamical localization for the Anderson model in various regimes.

Findings

SDL proven for $bZ^2$ and $bZ^3$ with Bernoulli distribution

Multiscale analysis ingredients lead to spectral and dynamical localization

Results extend to regimes with recent spectral analysis progress

Abstract

We prove that once one has the ingredients of a ``single-energy multiscale analysis (MSA) result'' on the $\mathbb{Z}^d$ lattice, several spectral and dynamical localization results can be derived, the most prominent being strong dynamical localization (SDL). In particular, given the recent progress at the bottom of the spectrum for the $\mathbb{Z}^2$ and $\mathbb{Z}^3$ cases with Bernoulli single site probability distribution, our results imply SDL in these regimes.