Uniform existence of the integrated density of states for models on $\ZZ^d$

TL;DR

This paper proves the existence of the integrated density of states for certain models on ^d, using an ergodic theorem that applies to various examples without relying on von Neumann algebras.

Contribution

It introduces a new ergodic theorem for Banach-space valued functions that ensures the existence of the integrated density of states in a broad setting.

Findings

Established convergence of the integrated density of states for finite-range operators

Provided explicit bounds on the speed of convergence

Applied results to periodic, percolation, and visible points models

Abstract

We provide an ergodic theorem for certain Banach-space valued functions on structures over $\ZZ^d$, which allow for existence of frequencies of finite patterns. As an application we obtain existence of the integrated density of states for associated finite-range operators in the sense of convergence of the distributions with respect to the supremum norm. These results apply to various examples including periodic operators, percolation models and nearest-neighbour hopping on the set of visible points. Our method gives explicit bounds on the speed of convergence in terms of the speed of convergence of the underlying frequencies. It uses neither von Neumann algebras nor a framework of random operators on a probability space.