{
  "id": "1809.02012",
  "version": "v1",
  "published": "2018-09-06T14:30:15.000Z",
  "updated": "2018-09-06T14:30:15.000Z",
  "title": "The peeling process on random planar maps coupled to an O(n) loop model (with an appendix by Linxiao Chen)",
  "authors": [
    "Timothy Budd"
  ],
  "comment": "61 pages, 18 figures",
  "categories": [
    "math.PR",
    "math-ph",
    "math.CO",
    "math.MP"
  ],
  "abstract": "We extend the peeling exploration introduced in arxiv:1506.01590 to the setting of Boltzmann planar maps coupled to a rigid $O(n)$ loop model. Its law is related to a class of discrete Markov processes obtained by confining random walks to the positive integers with a new type of boundary condition. As an application we give a rigorous justification of the phase diagram of the model presented in arXiv:1106.0153. This entails two results pertaining to the so-called fixed-point equation: the first asserts that any solution determines a well-defined model, while the second result, contributed by Chen in the appendix, establishes precise existence criteria. A scaling limit for the exploration process is identified in terms of a new class of positive self-similar Markov processes, going under the name of ricocheted stable processes. As an application we study distances on loop-decorated maps arising from a particular first passage percolation process on the maps. In the scaling limit these distances between the boundary and a marked point are related to exponential integrals of certain L\\'evy processes. The distributions of the latter can be identified in a fairly explicit form using machinery of positive self-similar Markov processes. Finally we observe a relation between the number of loops that surround a marked vertex in a Boltzmann loop-decorated map and the winding angle of a simple random walk on the square lattice. As a corollary we give a combinatorial proof of the fact that the total winding angle around the origin of a simple random walk started at $(p,p)$ and killed upon hitting $(0,0)$ normalized by $\\log p$ converges in distribution to a Cauchy random variable.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2018-09-06T14:30:15.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "random planar maps",
      "loop model",
      "linxiao chen",
      "peeling process",
      "positive self-similar markov processes"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 61,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}