Macroscopic loops in the loop O(n)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
Macroscopic loops in the loop $O(n)$ model at Nienhuis' critical point cover
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The loop O(n)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)O(n) model. It has been predicted by Nienhuis that for 0n20\le n\le 2, the loop O(n)O(n) model exhibits a phase transition at a critical parameter xc(n)=1/2+2nx_c(n)=1/\sqrt{2+\sqrt{2-n}}. For 0<n20 < n \le 2, the transition line has been further conjectured to separate a regime with short loops when x<xc(n)x < x_c(n) from a regime with macroscopic loops when xxc(n)x\ge x_c(n).

In this paper, we prove that for n[1,2]n\in [1,2] and x=xc(n)x=x_c(n), the loop O(n)O(n) model exhibits macroscopic loops. Apart from the case n=1n = 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 n1n \ge 1 and 0<x1n0 < 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 (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]n\in[1,2] and x=xc(n)x=x_c(n).

Cite this article

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

DOI 10.4171/JEMS/1012