Finitary codings for the random-cluster model and other infinite-range monotone models

TL;DR

This paper constructs finitary codings for the random-cluster model and related monotone models on graphs, showing exponential tail bounds in the subcritical regime and extending results to various models including Potts, loop O(n), and long-range Ising.

Contribution

It introduces a probabilistic method using coupling-from-the-past to create finitary codings for infinite-range monotone models, strengthening previous results and applying to multiple models.

Findings

Finitary coding exists when free and wired measures coincide.

Coding radius has exponential tails in the subcritical regime.

Constructs translation-equivariant codings on bZ^d with finite-valued i.i.d. processes.

Abstract

A random field $X = (X_v)_{v \in G}$ on a quasi-transitive graph $G$ is a factor of i.i.d. if it can be written as $X=\varphi(Y)$ for some i.i.d. process $Y= (Y_v)_{v \in G}$ and equivariant map $\varphi$. Such a map, also called a coding, is finitary if, for every vertex $v \in G$, there exists a finite (but random) set $U \subset G$ such that $X_v$ is determined by $\{Y_u\}_{u \in U}$. We construct a coding for the random-cluster model on $G$, and show that the coding is finitary whenever the free and wired measures coincide. This strengthens a result of H\"aggstr\"om--Jonasson--Lyons. We also prove that the coding radius has exponential tails in the subcritical regime. As a corollary, we obtain a similar coding for the subcritical Potts model. Our methods are probabilistic in nature, and at their heart lies the use of coupling-from-the-past for the Glauber dynamics. These methods apply to any monotone model satisfying mild technical (but natural) requirements. Beyond the random-cluster and Potts models, we describe two further applications -- the loop $O(n)$ model and long-range Ising models. In the case of $G = \mathbb{Z}^d$, we also construct finitary, translation-equivariant codings using a finite-valued i.i.d. process $Y$. To do this, we extend a mixing-time result of Martinelli--Olivieri to infinite-range monotone models on quasi-transitive graphs of sub-exponential growth.