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

This is the authors' abstract. We don't add key points for this paper.

The loop $O(n)$ model is a family of probability measures on collections of non-intersecting 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 < xc(n), and another with macroscopic loops when x ≥ xc(n), where xc(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 any (n,x) ∈ [1,1+δ) × (1- δ, 1], there are arbitrarily long loops surrounding typical faces in a finite domain. If n ∈ [1,1+δ) and x ∈ (1-δ,1/√n], we 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.