The loop O(n) model is a model for a random collection of non-intersecting loops on the hexagonal lattice, which is believed to be in the same universality class as the spin O(n) model. It has been predicted by Nienhuis that for 0 ≤ n ≤ 2, the loop O(n) model exhibits a phase transition at a critical parameter xc(n) = 1/ 2 + √ 2 − n. For 0 < n ≤ 2, the transition line has been further conjectured to separate a regime with short loops when x < xc(n) from a regime with macroscopic loops when x ≥ xc(n).In this paper, we prove that for n ∈ [1, 2] and x = xc(n), the loop O(n) model exhibits macroscopic loops. This is the first instance in which a loop O(n) model with n = 1 is shown to exhibit such behavior. A main tool in the proof is a new positive association (FKG) property shown to hold when n ≥ 1 and 0 < x ≤ 1 √ n . This property implies, using techniques recently developed for the random-cluster model, the following dichotomy: either long loops are exponentially unlikely or the origin is surrounded by loops at any scale (box-crossing property). We develop a 'domain gluing' technique which allows us to employ Smirnov's parafermionic observable to rule out the first alternative when n ∈ [1, 2] and x = xc(n).

Abstract. The loop O(n) model is a model for a random collection of non-intersecting loops on the hexagonal lattice, which is believed to be in the same universality class as the spin O(n) model. It has been conjectured that both the spin and the loop O(n) models exhibit exponential decay of correlations when n > 2. We verify this for the loop O(n) model with large parameter n, showing that long loops are exponentially unlikely to occur, uniformly in the edge weight x. Our proof provides further detail on the structure of typical configurations in this regime. Putting appropriate boundary conditions, when nx 6 is sufficiently small, the model is in a dilute, disordered phase in which each vertex is unlikely to be surrounded by any loops, whereas when nx 6 is sufficiently large, the model is in a dense, ordered phase which is a small perturbation of one of the three ground states.

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.

No abstract

No abstract