Lifshitz tails in the 3D Anderson model

TL;DR

This paper investigates the localization properties of the 3D Anderson model with weak disorder, establishing almost sure localization in a specific energy band and characterizing the correlation length behavior near the band edge.

Contribution

It provides a rigorous derivation of localization in the 3D Anderson model for small disorder and describes the correlation length's asymptotic behavior near the spectral edge.

Findings

Almost sure localization for energies below a certain threshold

Correlation length scales as the inverse square root of energy distance from the band edge

Completes an argument previously outlined in unpublished work

Abstract

Consider the 3D Anderson model with a zero mean and bounded i.i.d. random potential. Let $\lambda$ be the coupling constant measuring the strength of the disorder, and $\sigma(E)$ the self energy of the model at energy $E$. For any $\epsilon>0$ and sufficiently small $\lambda$, we derive almost sure localization in the band $E \le -\sigma(0)-\lambda^{4-\epsilon}$. In this energy region, we show that the typical correlation length $\xi_E$ behaves roughly as $O((|E|-\sigma(E))^{-1/2})$, completing the argument outlined in the unpublished work of T. Spencer.