{
  "id": "1702.02919",
  "version": "v1",
  "published": "2017-02-09T18:03:04.000Z",
  "updated": "2017-02-09T18:03:04.000Z",
  "title": "Connection probabilities for conformal loop ensembles",
  "authors": [
    "Jason Miller",
    "Wendelin Werner"
  ],
  "comment": "30 pages",
  "categories": [
    "math.PR",
    "math-ph",
    "math.MP"
  ],
  "abstract": "The goal of the present paper is to explain, based on properties of the conformal loop ensembles CLE$_\\kappa$ (both with simple and non-simple loops, i.e., for the whole range $\\kappa \\in (8/3, 8)$) how to derive the connection probabilities in conformal rectangles for a conditioned version of CLE$_\\kappa$ which can be interpreted as a CLE$_{\\kappa}$ with wired/free/wired/free boundary conditions on four boundary arcs (the wired parts being viewed as portions of to-be-completed loops). In particular, in the case of a conformal square, we prove that the probability that the two wired sides hook up so that they create one single loop is equal to $1/(1 - 2 \\cos (4 \\pi / \\kappa ))$. Comparing this with the corresponding connection probabilities for discrete O(N) models for instance indicates that if a dilute O(N) model (respectively a critical FK(q)-percolation model on the square lattice) has a non-trivial conformally invariant scaling limit, then necessarily this scaling limit is CLE$_\\kappa$ where $\\kappa$ is the value in $(8/3, 4]$ such that $-2 \\cos (4 \\pi / \\kappa )$ is equal to $N$ (resp. the value in $[4,8)$ such that $-2 \\cos (4\\pi / \\kappa)$ is equal to $\\sqrt {q}$). Our arguments and computations build on the one hand on Dub\\'edat's SLE commutation relations (as developed and used by Dub\\'edat, Zhan or Bauer-Bernard-Kyt\\\"ol\\\"a) and on the other hand, on the construction and properties of the conformal loop ensembles and their relation to Brownian loop-soups, restriction measures, and the Gaussian free field, as recently derived in works with Sheffield and with Qian.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2017-02-09T18:03:04.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "connection probabilities",
      "probability",
      "dubedats sle commutation relations",
      "conformal loop ensembles cle",
      "wired/free/wired/free boundary conditions"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 30,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}