A Banach space-valued ergodic theorem and the uniform approximation of the integrated density of states

TL;DR

This paper proves a Banach space-valued ergodic theorem for operators on infinite graphs, enabling the uniform approximation of the integrated density of states for various group structures and coloring schemes.

Contribution

It introduces a new ergodic theorem in Banach spaces that facilitates the uniform approximation of the IDS for operators on infinite graphs with equivariance conditions.

Findings

Established the existence of the IDS as a uniform limit of finite matrix approximants.

Provided explicit convergence estimates for the approximation process.

Applied the results to periodic and percolation-type Hamiltonians.

Abstract

In this paper we consider bounded operators on infinite graphs, in particular Cayley graphs of amenable groups. The operators satisfy an equivariance condition which is formulated in terms of a colouring of the vertex set of the underlying graph. In this setting it is natural to expect that the integrated density of states (IDS), or spectral distribution function, exists. We show that it can be defined as the uniform limit of approximants associated to finite matrices. The proof is based on a Banach space valued ergodic theorem which even allows explicit convergence estimates. Our result applies to a variety of group structures and colouring types, in particular to periodic operators and percolation-type Hamiltonians.