Localization Lengths and Boltzmann Limit for the Anderson Model at Small Disorders in Dimension 3

TL;DR

This paper establishes lower bounds on eigenfunction localization lengths in the 3D Anderson model at weak disorder and demonstrates that the macroscopic dynamics follow linear Boltzmann equations, extending previous results to higher dimensions.

Contribution

It provides the first lower bounds on localization lengths in 3D at small disorder and connects microscopic eigenfunction behavior to macroscopic Boltzmann dynamics.

Findings

Eigenfunctions have localization lengths at least on the order of λ^{-2}/log(1/λ).

Most eigenfunctions are localized with bounds depending on disorder strength.

Macroscopic lattice dynamics are governed by linear Boltzmann equations.

Abstract

We prove lower bounds on the localization length of eigenfunctions in the three-dimensional Anderson model at weak disorders. Our results are similar to those obtained by Schlag, Shubin and Wolff for dimensions one and two. We prove that with probability one, most eigenfunctions have localization lengths bounded from below by $O(\frac{\lambda^{-2}}{\log\frac1\lambda})$, where $\lambda$ is the disorder strength. This is achieved by time-dependent methods which generalize those developed by Erd\"os and Yau to the lattice and non-Gaussian case. In addition, we show that the macroscopic limit of the corresponding lattice random Schr\"odinger dynamics is governed by the linear Boltzmann equations.