H\"older continuity of the IDS for matrix-valued Anderson models

TL;DR

This paper establishes the H"older continuity of the Integrated Density of States for certain matrix-valued Anderson models, extending regularity results to higher dimensions and matrix sizes, with applications to specific models.

Contribution

It proves the regularity of the IDS for matrix-valued Anderson models in any dimension and matrix size, under certain group-theoretic assumptions, and introduces a new Thouless formula relating Lyapunov exponents to the IDS.

Findings

Existence of IDS for all dimensions and matrix sizes.

H"older continuity of IDS in 1D for arbitrary matrix sizes.

Application to specific models satisfying the group assumption.

Abstract

We study a class of continuous matrix-valued Anderson models acting on $L^{2}(\R^{d})\otimes \C^{N}$. We prove the existence of their Integrated Density of States for any $d\geq 1$ and $N\geq 1$. Then for $d=1$ and for arbitrary $N$, we prove the H\"older continuity of the Integrated Density of States under some assumption on the group $G_{\mu_{E}}$ generated by the transfer matrices associated to our models. This regularity result is based upon the analoguous regularity of the Lyapounov exponents associated to our model, and a new Thouless formula which relates the sum of the positive Lyapounov exponents to the Integrated Density of States. In the final section, we present an example of matrix-valued Anderson model for which we have already proved, in a previous article, that the assumption on the group $G_{\mu_{E}}$ is verified. Therefore the general results developed here can be applied to this model.