MACROSCOPIC LOOPS IN THE LOOP O(n) MODEL VIA THE XOR TRICK

The loop O(n) model is a family of probability measures on collections of nonintersecting loops on the hexagonal lattice, parameterized by a loop-weight n and an edge-weight x. Nienhuis predicts that, for 0 ≤ n ≤ 2, the model exhibits two regimes separated by (Formula presented): when x < x_c(n), the loop lengths have exponential tails, while when x ≥ x_c(n), the loops are macroscopic. In this paper, we prove three results regarding the existence of long loops in the loop O(n) model: – In the regime (n, x) ∈ [1, 1 + δ) × (1 − δ, 1] with δ > 0 small, a configuration sampled from a translation-invariant Gibbs measure will either contain an infinite path or have infinitely many loops surrounding every face. In the subregime n ∈ [1, 1 + δ) and (Formula presented), our results further imply Russo–Seymour–Welsh theory. This is the first proof of the existence of macroscopic loops in a positive area subset of the phase diagram. – Existence of loops whose diameter is comparable to that of a finite domain whenever n = 1, (Formula presented); this regime is equivalent to part of the antiferromagnetic regime of the Ising model on the triangular lattice. – Existence of noncontractible 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 planar 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.