Macroscopic loops in the loop 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
Macroscopic loops in the loop $O(n)$ model at Nienhuis' critical point cover
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The loop 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 model. It has been predicted by Nienhuis that for , the loop model exhibits a phase transition at a critical parameter . For , the transition line has been further conjectured to separate a regime with short loops when from a regime with macroscopic loops when .

In this paper, we prove that for and , the loop model exhibits macroscopic loops. Apart from the case , 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 and . 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 and .

Cite this article

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

DOI 10.4171/JEMS/1012