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 \(x_c(n)=\tfrac{1}{\sqrt{2+\sqrt{2-n}}}\). 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. This is the first instance in which a loop \(O(n)\) model with \(n\neq 1\) is shown to exhibit such behaviour. 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 (box-crossing property). We develop a 'domain gluing' technique which allows us to employ Smirnov's parafermionic observable to rule out the first alternative when \(x=x_c(n)\) and \(n\in[1,2]\).