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

### Hugo Duminil-Copin

Institut des Hautes Études Scientifiques, Bures-sur-Yvette, France and Université de Genève, Switzerland### Alexander Glazman

Universität Wien, Austria### Ron Peled

Tel Aviv University, Israel### Yinon Spinka

The University of British Columbia, Vancouver, Canada

## 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 $0≤n≤2$, the loop $O(n)$ model exhibits a phase transition at a critical parameter $x_{c}(n)=1/2+2−n $. For $0<n≤2$, 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 $x≥x_{c}(n)$.

In this paper, we prove that for $n∈[1,2]$ 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 $n≥1$ and $0<x≤n 1 $. 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 $n∈[1,2]$ and $x=x_{c}(n)$.

## Cite this article

Hugo Duminil-Copin, Alexander Glazman, Ron Peled, Yinon Spinka, Macroscopic loops in the loop $O(n)$ model at Nienhuis' critical point. J. Eur. Math. Soc. 23 (2021), no. 1, pp. 315–347

DOI 10.4171/JEMS/1012