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

Matan Harel, Yinon Spinka

Research output: Contribution to journalArticlepeer-review


A random field X = (Xv)v∈G on a quasi-transitive graph G is a factor of an i.i.d. process if it can be written as X = φ(Y) for some i.i.d. process Y = (Yv)v∈G and equivariant map φ. Such a map, also called a coding, is finitary if, for every vertex v ∈ G, there exists a finite (but random) set U ⊂ G such that Xv is determined by {Yu}u∈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 = Zd, 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.

Original languageEnglish
Article number51
JournalElectronic Journal of Probability
StatePublished - 2022
Externally publishedYes


  • coupling from the past
  • factor of iid
  • finitary coding
  • monotone specification
  • quasi-transitive graph
  • random-cluster model


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