## 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/^{p}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 ≤ 1/√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 n ∈ [1, 2] and x = x_{c}(n).

Original language | English |
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Pages (from-to) | 315-347 |

Number of pages | 33 |

Journal | Journal of the European Mathematical Society |

Volume | 23 |

Issue number | 1 |

DOIs | |

State | Published - 2021 |

## Keywords

- Conformal invariance
- Dichotomy theorem
- Dilute Potts model
- FKG inequality
- Kosterlitz-Thouless phase transition
- Loop O(n) model
- Macroscopic loops
- Parafermionic observable
- Russo-Seymour-Welsh theory
- Spin representation
- Two-dimensional critical phenomena