MACROSCOPIC LOOPS IN THE LOOP O(n) MODEL VIA THE XOR TRICK

• 成果类型: Article
• 署名作者: Crawford, Nicholas; Glazman, Alexander; Harel, Matan; Peled, Ron
• 署名单位: Technion Israel Institute of Technology; University of Innsbruck; Northeastern University; Tel Aviv University
• 刊物名称: ANNALS OF PROBABILITY
• ISSN/ISSBN: 0091-1798
• DOI: 10.1214/24-AOP1712
• 发表日期: 2025
• 页码: 478-508
• 关键词: random-cluster model; conformal-invariance; critical percolation; PHASE-TRANSITION; probabilities; point

摘要

The loop O(n) model is a family of probability measures on collections of nonintersecting loops on the hexagonal lattice, parameterized by a loop-weight n and an edge-weight x. Nienhuis predicts that, for 0 < n <2, the model exhibits two regimes separated by xc (n) = 1/2 + root 2 - n: when x < xc(n), the loop lengths have exponential tails, while when x >= xc(n), the loops are macroscopic. In this paper, we prove three results regarding the existence of long loops in the loop O(n) model: - In the regime (n,x) E [1,1 + delta) x (1 -delta, 1] with delta >0 small, a configuration sampled from a translation-invariant Gibbs measure will either contain an infinite path or have infinitely many loops surrounding every face. In the subregime n E [1, 1 + delta) and x E (1 - delta, 1/root n], our results further imply Russo-Seymour-Welsh theory. This is the first proof of the existence of macroscopic loops in a positive area subset of the phase diagram. - Existence of loops whose diameter is comparable to that of a finite do root main whenever n = 1, x E (1, 3]; this regime is equivalent to part of the antiferromagnetic regime of the Ising model on the triangular lattice. - Existence of noncontractible loops on a torus when n E [1, 2], x = 1. The main ingredients of the proof are: (i) the XOR trick: if omega is a collection of short loops and is a long loop, then the symmetric difference of omega and necessarily includes a long loop as well; (ii) a reduction of the problem of finding long loops to proving that a percolation process on an auxiliary planar graph, built using the Chayes-Machta and Edwards-Sokal geometric expansions, has no infinite connected components and (iii) a recent result on the percolation threshold of Benjamini-Schramm limits of planar graphs.