{
  "id": "1810.11302",
  "version": "v1",
  "published": "2018-10-26T12:59:37.000Z",
  "updated": "2018-10-26T12:59:37.000Z",
  "title": "Exponential decay in the loop $O(n)$ model: $n> 1$, $x<\\tfrac{1}{\\sqrt{3}}+\\varepsilon(n)$",
  "authors": [
    "Alexander Glazman",
    "Ioan Manolescu"
  ],
  "comment": "12 pages, 2 figures",
  "categories": [
    "math.PR",
    "math-ph",
    "math.MP"
  ],
  "abstract": "We show that the loop $O(n)$ model exhibits exponential decay of loop sizes whenever $n\\geq 1$ and $x<\\tfrac{1}{\\sqrt{3}}+\\varepsilon(n)$, for some suitable choice of $\\varepsilon(n)>0$. It is expected that, for $n \\leq 2$, the model exhibits a phase transition in terms of $x$, that separates regimes of polynomial and exponential decay of loop sizes. In this paradigm, our result implies that the phase transition for $n \\in (1,2]$ occurs at some critical parameter $x_c(n)$ strictly greater than that $x_c(1) = 1/\\sqrt3$. The value of the latter is known since the loop $O(1)$ model on the hexagonal lattice represents the contours of spin-clusters of the Ising model on the triangular lattice. The proof is based on developing $n$ as $1+(n-1)$ and exploiting the fact that, when $x<\\tfrac{1}{\\sqrt{3}}$, the Ising model exhibits exponential decay on any (possibly non simply-connected) domain. The latter follows from the positive association of the FK-Ising representation.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2018-10-26T12:59:37.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "60K35",
      "82B20"
    ],
    "keywords": [
      "exponential decay",
      "loop sizes",
      "phase transition",
      "hexagonal lattice represents",
      "ising model"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 12,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}