Macroscopic loops in the loop $O(n)$ model at Nienhuis' critical point

Abstract

The loop $O(n)$ model is a model for a random collection of non-intersecting loops on the hexagonal lattice, which is believed to be in the same universality class as the spin $O(n)$ model. It has been predicted by Nienhuis that for $0\le n\le 2$, the loop $O(n)$ model exhibits a phase transition at a critical parameter $x_c(n)=1/\sqrt{2+\sqrt{2-n}}$. For $0 < n \le 2$, the transition line has been further conjectured to separate a regime with short loops when $x < x_c(n)$ from a regime with macroscopic loops when $x\ge x_c(n)$.

In this paper, we prove that for $n\in [1,2]$ and $x=x_c(n)$, the loop $O(n)$ model exhibits macroscopic loops. Apart from the case $n = 1$, this constitutes the first regime of parameters for which macroscopic loops have been rigorously established. A main tool in the proof is a new positive association (FKG) property shown to hold when $n \ge 1$ and $0 < x \le\frac{1}{\sqrt{n}}$. This property implies, using techniques recently developed for the random-cluster model, the following dichotomy: either long loops are exponentially unlikely or the origin is surrounded by loops at any scale (boxcrossing property). We develop a “domain gluing” technique which allows us to employ Smirnov’s parafermionic observable to rule out the first alternative when $n\in[1,2]$ and $x=x_c(n)$.