Localization for Discrete One Dimensional Random Word Models

TL;DR

This paper proves spectral and dynamical localization for a class of one-dimensional random Schrödinger operators with potentials formed by concatenating random words, including models with local correlations like the random dimer model.

Contribution

It introduces a framework for analyzing localization in models with correlated potentials using scattering theory and Furstenberg's theorem.

Findings

Spectral localization established for the models.

Dynamical localization proven away from a finite set of energies.

Applicable to models with local correlations such as random dimer and polymer models.

Abstract

We consider Schr\"odinger operators in $\ell^2(\Z)$ whose potentials are obtained by randomly concatenating words from an underlying set $\mathcal{W}$ according to some probability measure $\nu$ on $\mathcal{W}$. Our assumptions allow us to consider models with local correlations, such as the random dimer model or, more generally, random polymer models. We prove spectral localization and, away from a finite set of exceptional energies, dynamical localization for such models. These results are obtained by employing scattering theoretic methods together with Furstenberg's theorem to verify the necessary input to perform a multiscale analysis.