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 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 $0 \leq n \leq 2$, the model exhibits two regimes: one with short loops when $x < x_c(n)$, and another with macroscopic loops when $x \geq x_c(n)$, where $x_c(n) = 1/\sqrt{2 + \sqrt{2-n}}$. 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 $\delta >0$ and any $(n,x) \in [1,1+\delta) \times (1- \delta, 1]$, there are arbitrarily long loops surrounding typical faces in a finite domain. If $n \in [1,1+\delta)$ and $x \in (1-\delta,1/\sqrt{n}]$, 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 $n=1, x \in (1,\sqrt{3}]$; 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 $n \in [1,2], x=1$. The main ingredients of the proof are: (i) the `XOR trick': if $\omega$ is a collection of short loops and $\Gamma$ is a long loop, then the symmetric difference of $\omega$ and $\Gamma$ 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.