A lower bound for the density of states of the lattice Anderson model

TL;DR

This paper establishes a positive lower bound on the density of states for the multi-dimensional lattice Anderson model under specific conditions on the randomness distribution.

Contribution

It provides the first rigorous proof of a positive lower bound on the density of states for the Anderson model with certain distribution assumptions.

Findings

Density of states is strictly positive for almost every energy in the spectrum.

Lower bound depends on the distribution's properties.

Results apply to models with bounded, compactly supported distributions.

Abstract

We consider the Anderson model on the multi-dimensional cubic lattice and prove a positive lower bound on the density of states under certain conditions. For example, if the random variables are independently and identically distributed and the probability measure has a bounded Lebesgue density with compact support, and if this density is essentially bounded away from zero on its support, then we prove that the density of states is strictly positive for Lebesgue-almost every energy in the deterministic spectrum.