{
  "id": "1707.09335",
  "version": "v1",
  "published": "2017-07-28T17:23:23.000Z",
  "updated": "2017-07-28T17:23:23.000Z",
  "title": "Macroscopic loops in the loop $O(n)$ model at Nienhuis' critical point",
  "authors": [
    "Hugo Duminil-Copin",
    "Alexander Glazman",
    "Ron Peled",
    "Yinon Spinka"
  ],
  "comment": "26 pages, 8 figures",
  "categories": [
    "math.PR",
    "math-ph",
    "math.MP"
  ],
  "abstract": "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 $0\\le n\\le 2$ the loop $O(n)$ model exhibits a phase transition at a critical parameter $x_c(n)=\\tfrac{1}{\\sqrt{2+\\sqrt{2-n}}}$. For $0<n\\le 2$, the transition line has been further conjectured to separate a regime with short loops when $x<x_c(n)$ from a regime with macroscopic loops when $x\\ge x_c(n)$. In this paper, we prove that for $n\\in [1,2]$ and $x=x_c(n)$ the loop $O(n)$ model exhibits macroscopic loops. This is the first instance in which a loop $O(n)$ model with $n\\neq 1$ is shown to exhibit such behaviour. A main tool in the proof is a new positive association (FKG) property shown to hold when $n \\ge 1$ and $0<x\\le\\frac{1}{\\sqrt{n}}$. 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 (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 $x=x_c(n)$ and $n\\in[1,2]$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2017-07-28T17:23:23.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "60K35",
      "82B20",
      "82B27"
    ],
    "keywords": [
      "macroscopic loops",
      "critical point",
      "employ smirnovs parafermionic observable",
      "first instance",
      "hexagonal lattice"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 26,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}