{
  "id": "1704.01950",
  "version": "v1",
  "published": "2017-04-06T17:37:49.000Z",
  "updated": "2017-04-06T17:37:49.000Z",
  "title": "Limits of the boundary of random planar maps",
  "authors": [
    "Loïc Richier"
  ],
  "comment": "45 pages",
  "categories": [
    "math.PR"
  ],
  "abstract": "We discuss asymptotics for the boundary of critical Boltzmann planar maps under the assumption that the distribution of the degree of a typical face is in the domain of attraction of a stable distribution with parameter $\\alpha \\in (1,2)$. First, in the dense phase corresponding to $\\alpha\\in(1,3/2)$, we prove that the scaling limit of the boundary is the random stable looptree with parameter $(\\alpha-1/2)^{-1}$. Second, we show the existence of a phase transition through local limits of the boundary: in the dense phase, the boundary is tree-like, while in the dilute phase corresponding to $\\alpha\\in(3/2,2)$, it has a component homeomorphic to the half-plane. As an application, we identify the limits of loops conditioned to be large in the rigid $O(n)$ loop model on quadrangulations, proving thereby a conjecture of Curien and Kortchemski.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2017-04-06T17:37:49.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "random planar maps",
      "dense phase",
      "critical boltzmann planar maps",
      "loop model",
      "dilute phase"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 45,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}