Spectral Properties of the Discrete Random Displacement Model

TL;DR

This paper studies the spectral characteristics of a discrete random displacement model, revealing how randomness affects the spectrum edges and the density of states, especially in one-dimensional Bernoulli cases.

Contribution

It characterizes the spectrum edges of the model and analyzes the density of states singularity in a one-dimensional Bernoulli setting.

Findings

Spectrum edges are explicitly characterized.

Integrated density of states exhibits a $1/ ext{log}^2$-singularity.

Results apply to both external and internal band edges.

Abstract

We investigate spectral properties of a discrete random displacement model, a Schr\"odinger operator on $\ell^2(\Z^d)$ with potential generated by randomly displacing finitely supported single-site terms from the points of a sublattice of $\Z^d$. In particular, we characterize the upper and lower edges of the almost sure spectrum. For a one-dimensional model with Bernoulli distributed displacements, we can show that the integrated density of states has a $1/\log^2$-singularity at external as well as internal band edges.