Approximation of the integrated density of states on sofic groups

TL;DR

This paper demonstrates that the integrated density of states for self-adjoint operators on sofic groups can be approximated by finite volume models, applicable in both deterministic and random contexts, including various group types.

Contribution

It introduces a general framework for approximating the integrated density of states on sofic groups without relying on ergodic theorems, covering a wide class of operators and applications.

Findings

Approximation of the integrated density of states via finite models.

Applicability to both deterministic and random operators.

Coverage of diverse group structures including trees and lattices.

Abstract

In this paper we study spectral properties of self-adjoint operators on a large class of geometries given via sofic groups. We prove that the associated integrated densities of states can be approximated via finite volume analogues. This is investigated in the deterministic as well as in the random setting. In both cases we cover a wide range of operators including in particular unbounded ones. The large generality of our setting allows to treat applications from long-range percolation and the Anderson model. Our results apply to operators on Z^d, amenable groups, residually finite groups and therefore in particular to operators on trees. All convergence results are established without any ergodic theorem at hand.