{
  "id": "1012.3726",
  "version": "v2",
  "published": "2010-12-16T19:37:52.000Z",
  "updated": "2012-10-10T13:40:30.000Z",
  "title": "The topology of scaling limits of positive genus random quadrangulations",
  "authors": [
    "Jérémie Bettinelli"
  ],
  "comment": "Published in at http://dx.doi.org/10.1214/11-AOP675 the Annals of Probability (http://www.imstat.org/aop/) by the Institute of Mathematical Statistics (http://www.imstat.org)",
  "journal": "Annals of Probability 2012, Vol. 40, No. 5, 1897-1944",
  "doi": "10.1214/11-AOP675",
  "categories": [
    "math.PR"
  ],
  "abstract": "We discuss scaling limits of large bipartite quadrangulations of positive genus. For a given g, we consider, for every $n\\ge1$, a random quadrangulation $\\mathfrak{q}_n$ uniformly distributed over the set of all rooted bipartite quadrangulations of genus g with n faces. We view it as a metric space by endowing its set of vertices with the graph metric. As n tends to infinity, this metric space, with distances rescaled by the factor $n^{-1/4}$, converges in distribution, at least along some subsequence, toward a limiting random metric space. This convergence holds in the sense of the Gromov-Hausdorff topology on compact metric spaces. We show that, regardless of the choice of the subsequence, the limiting space is almost surely homeomorphic to the genus g-torus.",
  "revisions": [
    {
      "version": "v2",
      "updated": "2012-10-10T13:40:30.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "positive genus random quadrangulations",
      "scaling limits",
      "large bipartite quadrangulations",
      "limiting random metric space",
      "compact metric spaces"
    ],
    "tags": [
      "journal article"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2010arXiv1012.3726B"
    }
  }
}