Smoothness of integrated density of states and level statistics of the Anderson model when single site distribution is convolution with the Cauchy distribution

TL;DR

This paper studies the Anderson model with a specific type of disorder distribution, proving the smoothness of the integrated density of states and analyzing eigenvalue statistics without localization assumptions.

Contribution

It demonstrates the infinite differentiability of the IDS for a convolution of Cauchy and arbitrary measures and explores eigenvalue statistics in higher dimensions without localization.

Findings

IDS is infinitely differentiable regardless of disorder strength

Eigenvalue statistics analyzed without assuming localization

Results apply to models with convolution of Cauchy distribution

Abstract

In this work we consider the Anderson model on $\ell^2(\mathbb{Z}^d)$ when the single site distribution (SSD) is given by $\mu_1 * \mu_2$, where $\mu_1$ is the Cauchy distribution and $\mu_2$ is any probability measure. For this model we prove that the integrated density of states (IDS) is infinitely differentiable irrespective of the disorder strength. Also, we investigate the local eigenvalue statistics of this model in $d\ge 2$, without any assumption on the localization property.