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

Abstract

A random field $X={(X_v)}_{v\in G}$ on a quasi-transitive graph G is a factor of an i.i.d. process if it can be written as $X=\mathit{\phi }(Y)$ for some i.i.d. process $Y={(Y_v)}_{v\in G}$ and equivariant map φ. 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 Häggström–Jonasson–Lyons [18]. 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 [22] to infinite-range monotone models on quasi-transitive graphs of sub-exponential growth.