Uniform existence of the IDS on lattices and groups

TL;DR

This paper develops a general framework for thermodynamic limits in lattice and group models, establishing criteria for uniform convergence, exemplified by eigenvalue counting functions converging uniformly to the integrated density of states.

Contribution

It introduces a unified approach to thermodynamic limits with conditions ensuring uniform convergence across parameters, applied to spectral analysis.

Findings

Criteria for uniform thermodynamic limits established

Eigenvalue counting functions converge uniformly to the IDS

Framework applicable to various lattice and group models

Abstract

We present a general framework for thermodynamic limits and its applications to a variety of models. In particular we will identify criteria such that the limits are uniform in a parameter. All results are illustrated with the example of eigenvalue counting functions converging to the integrated density of states. In this case, the convergence is uniform in the energy.