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

Miguel Donoso-Echenique, Felix Pogorzelski, Michael Schrödl-Baumann

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 Lück 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.

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

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 Lück 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- limit of periodic measures admits an approximation in the local and empirical sense.
Paper Structure (21 sections, 25 theorems, 128 equations)

This paper contains 21 sections, 25 theorems, 128 equations.

Table of Contents

  1. Introduction and main results
  2. Background of results
  3. Description of results
  4. Preliminaries
  5. Graphs
  6. Cayley graphs
  7. The shift action and $G$-processes
  8. Operators
  9. Spectral calculus
  10. Sofic groups and local and empirical convergence
  11. Sofic groups and approximations
  12. Local and empirical convergence
  13. Periodic measures and local empirical convergence on sofic groups
  14. Approximation by periodic measures
  15. Local and empirical convergence for PA groups or quotients
  16. ...and 6 more sections

Key Result

Proposition 7

Let $\mu \in \mathrm{Prob}(\mathcal{X}^G,G)$ and $(\mu_n)$ be a sequence with $\mu_n \in \mathrm{Prob}(\mathcal{X}^{V_n})$ for a sofic approximation $(V_n,\sigma_n)$. If $\mu$ is ergodic, then

Theorems & Definitions (73)

  • Definition 1: Sofic groups
  • Definition 2
  • Definition 3
  • Definition 4: lw$^{*}$-convergence and le-convergence
  • Remark 5
  • Definition 6: Sofic and residually finite models
  • Proposition 7: Aus16*Corollary 5.7
  • Remark 8
  • Lemma 9
  • proof
  • ...and 63 more