Continuity of integrated density of states -- independent randomness

TL;DR

This paper investigates the continuity of the integrated density of states in random models with independent randomness, establishing that certain regularity conditions on the single site distribution imply similar regularity for the density of states.

Contribution

It demonstrates that for models with independent randomness, the regularity of the single site distribution directly determines the regularity of the integrated density of states.

Findings

Density of states inherits the Hölder continuity of the single site distribution.

Results apply to Anderson models with various background structures.

Continuity properties are valid for models with arbitrary free parts.

Abstract

In this paper we discuss the continuity properties of the integrated density of states for random models based on that of the single site distribution. Our results are valid for models with independent randomness with arbitrary free parts. In particular in the case of the Anderson type models (with stationary, growing, decaying randomness) on the $\nu$ dimensional lattice, with or without periodic and almost periodic backgrounds, we show that if the single site distribution is uniformly $\alpha$-H\"older continuous, $ 0 < \alpha \leq 1$, then the density of states is also uniformly $\alpha$-H\"older continuous.