Discrete Schr\"odinger operators with random alloy-type potential

TL;DR

This paper reviews recent advances in understanding localization phenomena for discrete alloy-type Schrödinger operators with random potentials, highlighting the use of multiscale analysis and fractional moment methods.

Contribution

It summarizes new results on localization for these models, especially addressing challenges posed by non-monotone dependence on random variables.

Findings

Localization established for discrete alloy-type models

Application of multiscale analysis and fractional moment methods

Handling non-monotone potential dependencies

Abstract

We review recent results on localization for discrete alloy-type models based on the multiscale analysis and the fractional moment method, respectively. The discrete alloy-type model is a family of Schr\"odinger operators $H_\omega = - \Delta + V_\omega$ on $\ell^2 (\ZZ^d)$ where $\Delta$ is the discrete Laplacian and $V_\omega$ the multiplication by the function $V_\omega (x) = \sum_{k \in \ZZ^d} \omega_k u(x-k)$. Here $\omega_k$, $k \in \ZZ^d$, are i.i.d. random variables and $u \in \ell^1 (\ZZ^d ; \RR)$ is a so-called single-site potential. Since $u$ may change sign, certain properties of $H_\omega$ depend in a non-monotone way on the random parameters $\omega_k$. This requires new methods at certain stages of the localization proof.