Spectral measures of factor of i.i.d. processes on vertex-transitive graphs

TL;DR

This paper characterizes the spectral measures of factor of i.i.d. processes on vertex-transitive graphs, establishing conditions for when a measure arises from such processes and relating them to spectral measures of the graph.

Contribution

It provides a complete characterization of spectral measures of factor of i.i.d. processes on vertex-transitive graphs and links them to spectral measures of the graph itself.

Findings

Spectral measure of a factor of i.i.d. process is absolutely continuous w.r.t. the graph's spectral measure.

The set of spectral measures of factor of i.i.d. processes equals the set of $ar d_2$-limits of such processes.

Characterization applies to infinite vertex-transitive graphs.

Abstract

We prove that a measure on $[-d, d]$ is the spectral measure of a factor of i.i.d. process on a vertex-transitive infinite graph if and only if it is absolutely continuous with respect to the spectral measure of the graph. Moreover, we show that the set of spectral measures of factor of i.i.d. processes and that of $\bar d_2$-limits of factor of i.i.d. processes are the same.