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

Abstract

Let G be a quasi-transitive graph on V. A random field X = (Xv) v∈V whose distribution is invariant under all automorphisms of G is said to be a factor of i.i.d. if there exists an i.i.d. process Y = (Yv) v∈V and an equivariant map ϕ such that ϕ(Y ) has the same distribution as X. Such a map, also called a coding, is said to be finitary if, for every v ∈ V, there exists a finite (but random) set U ⊂ V such that ϕ(Y )v is determined by {Yu} u∈U . We construct a coding for the random-cluster model on general quasi-transitive graphs, 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 sub-critical regime. As a corollary, we obtain a finitary coding for the sub-critical Potts model on G whose coding radius has exponential tails. In the case of G = Z d , we also construct a finitary, translation-equivariant coding for the sub-critical random-cluster and Potts models using a finite-valued i.i.d. process Y . To do this, we extend a mixing-time result of to infinite-range monotone models on quasi-transitive graphs of sub-exponential growth. Our methods also apply to any monotone model satisfying mild technical (but natural) requirements.

Finitely dependent processes are finitary

“…. , F s−1 ⊂ F s , it follows that ρ(u , w) > s. Using (7), we obtain The above proposition established the existence of a finitary cell process A. In particular, A n is an invariant set which has high density when n is large.…”

“…A total order (also with the order type of Q) which is a finitary factor (with exponential tails on the coding radius) of an i.i.d. process with finite entropy was constructed in [7] for any quasi-transitive graph satisfying a geometric condition similar to (1). The application in [7] did not require the i.i.d.…”

Low-temperature Ising dynamics with random initializations

“…For general q, [5] showed that the random-cluster dynamics on subsets of Z 2 mixes in OpN log N q time at all low temperatures (see also [3] where different boundary conditions were considered). More generally, the classical proof of [35] (shown to readily extend to random-cluster dynamics in [21]) implies fast mixing of the random-cluster dynamics on pZ{nZq d whenever the randomcluster model has a WSM property; recall that this property was shown to hold at all low temperatures in all dimensions when q " 2 in [14].…”

“…These include the absence of global monotonicity of the restricted dynamics, and the fact that there is no minimal configuration within a phase (only the distribution p π). While [35] (and extensions such as [21]) recurse over the probability of disagreement between two Markov chains initialized from the maximal and minimal configurations respectively, in our setting the minimal configuration lies outside the plus phase.…”

Sampling from Potts on random graphs of unbounded degree via random-cluster dynamics

“…As such, any sampling algorithm for the random-cluster model yields one for the ferromagnetic Potts model with essentially no computational overhead. This fact has led to significantly improved sampling algorithms for the Potts model in various low-temperature settings [2,3,7,8,12,13,20,34,39,41] and more generally, to a broad interest in dynamics for the random-cluster model [6,9,[21][22][23][24]26,28].…”