Localization for alloy-type models with non-monotone potentials

TL;DR

This paper advances the understanding of localization phenomena in alloy-type models with non-monotone potentials by refining multiscale analysis and fractional moment methods, establishing exponential localization and a Wegner estimate for certain classes of potentials.

Contribution

It develops new fractional moment and multiscale analysis techniques for alloy-type models with sign-changing potentials, proving localization and deriving a Wegner estimate.

Findings

Proves exponential localization for discrete alloy-type models with finite support and fixed sign at boundary.

Establishes a Wegner estimate for exponentially decaying, non-finitely supported potentials.

Refines fractional moment method for non-monotone potentials.

Abstract

We consider a family of self-adjoint operators [H_\omega = - \Delta + \lambda V_\omega, \quad \omega \in \Omega = \bigtimes_{k \in \ZZ^d} \RR,] on the Hilbert space $\ell^2 (\ZZ^d)$ or $L^2 (\RR^d)$. Here $\Delta$ denotes the Laplace operator (discrete or continuous), $V_\omega$ is a multiplication operator given by the function $$V_\omega (x) = \sum_{k \in \ZZ^d} \omega_k u(x-k) on $\ZZ^d$, or \quad V_\omega (x) = \sum_{k \in \ZZ^d} \omega_k U(x-k) on $\RR^d$,$$ and $\lambda > 0$ is a real parameter modeling the strength of the disorder present in the model. The functions $u:\ZZ^d \to \RR$ and $U:\RR^d \to \RR$ are called single-site potential. Moreover, there is a probability measure $\PP$ on $\Omega$ modeling the distribution of the individual configurations $\omega \in \Omega$. The measure $\PP = \prod_{k \in \ZZ^d} \mu$ is a product measure where $\mu$ is some probability measure on $\RR$ satisfying certain regularity assumptions. The operator on $L^2 (\RR^d)$ is called alloy-type model, and its analogue on $\ell^2 (\ZZ^d)$ discrete alloy-type model. This thesis refines the methods of multiscale analysis and fractional moments in the case where the single-site potential is allowed to change its sign. In particular, we develop the fractional moment method and prove exponential localization for the discrete alloy-type model in the case where the support of $u$ is finite and $u$ has fixed sign at the boundary of its support. We also prove a Wegner estimate for the discrete alloy-type model in the case of exponentially decaying but not necessarily finitely supported single-site potentials. This Wegner estimate is applicable for a proof of localization via multiscale analysis.