Random operators, spectral measures, and local empirical convergence in sofic groups

TL;DR

This paper studies how to approximate spectral distributions of random operators on sofic groups using locally and empirically converging measures, establishing various convergence results including a Lück type theorem.

Contribution

It introduces new approximation theorems for spectral measures of random operators on sofic groups, extending to complex coefficients and general invariant measures.

Findings

Weak convergence of density of states measures established.

Pointwise convergence of spectral measures proved for rational-valued operators.

Applicable to periodically approximable groups and invariant measures as weak-* limits.

Abstract

In this paper, we consider the problem of approximating the spectral distribution for a class of random operators over sofic groups. For this purpose, we make use of the concept of locally and empirically converging measures defined by Austin. We establish weak convergence of the density of states measures along random finite-volume analogs. For operators taking finitely many rational values, we prove a L\"uck type approximation theorem yielding pointwise convergence of the spectral measures. In the wider context of arbitrary complex coefficients, we show pointwise convergence of the spectral distribution functions along adapted approximants with varying rational coefficients. Our results apply to the class of periodically approximable groups as defined by Bowen. More generally, we show that every invariant probability measure on a finite-state configuration space that arises as a weak-$\ast$ limit of periodic measures admits an approximation in the local and empirical sense.