{
  "id": "2001.11977",
  "version": "v1",
  "published": "2020-01-31T17:50:59.000Z",
  "updated": "2020-01-31T17:50:59.000Z",
  "title": "Macroscopic loops in the loop~$O(n)$ model via the XOR trick",
  "authors": [
    "Nicholas Crawford",
    "Alexander Glazman",
    "Matan Harel",
    "Ron Peled"
  ],
  "comment": "32 pages, 4 figures",
  "categories": [
    "math.PR",
    "math-ph",
    "math.MP"
  ],
  "abstract": "The loop $O(n)$ model is a family of probability measures on collections of non-intersecting loops on the hexagonal lattice, parameterized by $(n,x)$, where $n$ is a loop weight and $x$ is an edge weight. Nienhuis predicts that, for $0 \\leq n \\leq 2$, the model exhibits two regimes: one with short loops when $x < x_c(n)$, and another with macroscopic loops when $x \\geq x_c(n)$, where $x_c(n) = 1/\\sqrt{2 + \\sqrt{2-n}}$. In this paper, we prove three results regarding the existence of long loops in the loop $O(n)$ model. Specifically, we show that, for some $\\delta >0$ and any $(n,x) \\in [1,1+\\delta) \\times (1- \\delta, 1]$, there are arbitrarily long loops surrounding typical faces in a finite domain. If $n \\in [1,1+\\delta)$ and $x \\in (1-\\delta,1/\\sqrt{n}]$, we can conclude the loops are macroscopic. Next, we prove the existence of loops whose diameter is comparable to that of a finite domain whenever $n=1, x \\in (1,\\sqrt{3}]$; this regime is equivalent to part of the antiferromagnetic regime of the Ising model on the triangular lattice. Finally, we show the existence of non-contractible loops on a torus when $n \\in [1,2], x=1$. The main ingredients of the proof are: (i) the `XOR trick': if $\\omega$ is a collection of short loops and $\\Gamma$ is a long loop, then the symmetric difference of $\\omega$ and $\\Gamma$ 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 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.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2020-01-31T17:50:59.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "60K35",
      "82B05",
      "82B20",
      "82B26",
      "82B27",
      "60C05"
    ],
    "keywords": [
      "macroscopic loops",
      "xor trick",
      "short loops",
      "finite domain",
      "long loops surrounding typical faces"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 32,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}