Macroscopic loops in the loop~$O(n)$ model via the XOR trick

Abstract

The loop O(n) model is a family of probability measures on collections of nonintersecting loops on the hexagonal lattice, parameterized by (n, x), where n is a loop weight and x is an edge weight. Nienhuis predicts that, for 0 ≤ n ≤ 2, the model exhibits two regimes: one with short loops when x < x c (n), and another with macroscopic loops when x ≥ x c (n), where x c (n) = 1 2 + √ 2 − n. In this paper, we prove three results regarding the existence of long loops in the loop O(n) model. Specifically, we show that, for some δ > 0 and anywe can conclude the loops are macroscopic. Next, we prove the existence of loops whose diameter is comparable to that of a finite domain whenever n = 1, x ∈ (1, √ 3]; this regime is equivalent to part of the antiferromagnetic regime of the Ising model on the triangular lattice. Finally, we show the existence of non-contractible loops on a torus when n ∈ [1, 2], x = 1.The main ingredients of the proof are: (i) the 'XOR trick': if ω is a collection of short loops and Γ is a long loop, then the symmetric difference of ω and Γ necessarily includes a long loop as well; (ii) a reduction of the problem of finding long loops to proving that a percolation process on an auxiliary graph, built using the Chayes-Machta and Edwards-Sokal geometric expansions, has no infinite connected components; and (iii) a recent result on the percolation threshold of Benjamini-Schramm limits of planar graphs.

Structure of Gibbs measures for planar FK-percolation and Potts models

“…At the moment, the phase diagram is understood only in several regions of parameters. For recent results on the two types of behaviour, we direct the reader to the recent works [10,16,28,27].…”