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 non-intersecting loops on the hexagonal lattice, parameterized by a loop-weight $n$ and an edge-weight $x$. Nienhuis predicts that, for $0 \leq n \leq 2$, the model exhibits two regimes separated by $x_c(n) = 1/\sqrt{2 + \sqrt{2-n}}$: when $x < x_c(n)$, the loop lengths have exponential tails, while, when $x \geq x_c(n)$, 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 $(n,x) \in [1,1+δ) \times (1- δ, 1]$ 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 $n \in [1,1+δ)$ and $x \in (1-δ,1/\sqrt{n}]$ 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 $n=1, x \in (1,\sqrt{3}]$; this regime is equivalent to part of the antiferromagnetic regime of the Ising model on the triangular lattice. - 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 $ω$ 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.

Figures (4)

• Figure 1: Samples of the loop $O(n)$ model for two nearby parameter values: the first in the predicted exponential decay regime and the second in the predicted macroscopic loops regime. The longest loops are colored for visibility.
• Figure 2: The predicted phase diagram for the loop $O(n)$ model. The critical line $x_c$ separating the regime of exponential decay from the regime of macroscopic loops is plotted in bold. The region $n\geq 1, x\leq \tfrac{1}{\sqrt{n}}$ where a dichotomy between the two behaviors is proved is denoted FKG region. Orange lines show regions where exponential decay is proved. Red dots or lines mark regions where macroscopic loops are proven to occur. The results of this paper are indicated in blue: macroscopic loops are established in the filled area, while exponential decay is ruled out in the dashed area and segments. Picture adapted from GlaMan21b.
• Figure 3: Left: a sample of of the loop $O(1)$ model with $x=1$ and empty boundary conditions, with the longest loop in red. Middle: a large loop superimposed on the configuration. Right: the result of the XOR operation between the configuration and the loop. Again, the longest loop is in red. Note that parts of the orange and blue loops are cut into smaller loops by the operation.
• Figure 4: The construction of the planar graph $G$. Left: A loop configuration $\omega$. Its intersection with $\mathcal{D}$ is shown in bold and the edges outside are dotted. Right: The graph $G$ associated with $\omega$. To construct it, we associate a vertex to each loop of $\omega$ (big black circles), edge of $\mathcal{D}$ disjoint with $\omega$ (blue dots), and face of $\mathcal{D}$ (green hexagons). We add an edge between two vertices associated with a face and a bordering edge or loop, or two vertices associated with two edges sharing an endpoint or an edge and a loop sharing an element of $\mathcal{D}$.

Theorems & Definitions (52)

• Theorem 1
• Corollary 1.1
• Corollary 1.2
• Theorem 2
• Corollary 1.3
• Corollary 1.4
• Theorem 3
• Lemma 1.5
• proof
• Proposition 1.6