Factor of iid's through stochastic domination

TL;DR

This paper introduces a new method to establish that certain percolation processes on amenable graphs are factors of iid, leveraging stochastic domination and monotone limits, with implications for USF and Ising models.

Contribution

It provides a novel approach to prove factor of iid properties for percolation processes, including USF and Ising models, on amenable graphs.

Findings

USF is a factor of iid for recurrent graphs

USF is a finite-valued finitary fiid on amenable graphs

Critical Ising model on Z^d is a finite-valued finitary fiid

Abstract

We develop a method to prove that certain percolation processes on amenable random rooted graphs are factors of iid (fiid), given that the process is a monotone limit of random finite subgraphs that satisfy a certain independent stochastic domination property. Among the consequences are the previously open claims that the Uniform Spanning Forest (USF) is a factor of iid for recurrent graphs, it is a finite-valued finitary fiid on amenable graphs, and that the critical Ising model on $\Z^d$ is a finite-valued finitary fiid, using the known uniqueness of the Gibbs measure.