Finitary codings for gradient models and a new graphical representation for the six-vertex model

Abstract

It is known that the Ising model on Z d at a given temperature is a finitary factor of an i.i.d. process if and only if the temperature is at least the critical temperature. Below the critical temperature, the plus and minus states of the Ising model are distinct and differ from one another by a global flip of the spins. We show that it is only this global information which poses an obstruction for being finitary by showing that the gradient of the Ising model is a finitary factor of i.i.d. at all temperatures. As a consequence, we deduce a volume-order large deviation estimate for the energy. A similar result is shown for the Potts model.A result in the same spirit is also shown for the six-vertex model, which is itself the gradient of a height function, with parameter c 6.4. We show that the gradient of the height function is not a finitary factor of an i.i.d. process, but that its "Laplacian" is. For this, we introduce a coupling between the six-vertex model with c ≥ 2 and a new graphical representation of it, reminiscent of the Edwards-Sokal coupling between the Potts and random-cluster models. We believe that this graphical representation may be of independent interest and could serve as a tool in further understanding of the six-vertex model.To provide further support for the ubiquity of this type of phenomenon, we also prove an analogous result for the so-called beach model.The tools and techniques used in this paper are probabilistic in nature. The heart of the argument is to devise a suitable tree structure on the clusters of the underlying percolation process (associated to the graphical representation of the given model), which can be revealed piece-by-piece via exploration.

Spins, percolation and height functions

“…We obtain it for ω and ω when a + b ≤ 1, and for σ and σ for all a, b ∈ (0, 1]. These are generalizations of the results of [19,37]. • In Sect.…”

“…Remark 4. We note that the left-hand side of the table above for q = q = 2 and a = b describes the process studied by Ray and Spinka [37].…”

“…For instance, a useful feature of this coupling will be that for a + b ≥ 1, the two percolation configurations (ω, ω ) are such that no edge and its dual edge are simultaneously open. This does not hold when a + b ≤ 1 which is the regime studied in [37].…”

“…As a result, we give a unified framework for some of the known relations between these models [1,6,9,15,19,23,30,31,35,37,39,40].…”

“…We note that in the special case q = q = 2 and a = b ≤ 1/2 the coupled percolation models (ω, ω ) studied here were independently introduced by Ray and Spinka in the article [37] which appeared during the preparation of this manuscript. For q = q = 2, the laws of the marginals on ω and ω are also present in the work of Glazman and Peled [19] on the six-vertex model, and are closely related to the percolation models introduced by Pfister and Velenik [35].…”