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

Hugo Duminil-Copin, Alexander Glazman, Ron Peled, Yinon Spinka

Research output: Contribution to journalArticlepeer-review


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 xc(n) = 1/p2 + √2 − n. For 0 < n ≤ 2, the transition line has been further conjectured to separate a regime with short loops when x < xc(n) from a regime with macroscopic loops when x ≥ xc(n). In this paper, we prove that for n ∈ [1, 2] and x = xc(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 = xc(n).

Original languageEnglish
Pages (from-to)315-347
Number of pages33
JournalJournal of the European Mathematical Society
Issue number1
StatePublished - 2021


  • 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


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