{
  "id": "1308.1807",
  "version": "v2",
  "published": "2013-08-08T10:23:18.000Z",
  "updated": "2014-08-19T10:45:02.000Z",
  "title": "A glimpse of the conformal structure of random planar maps",
  "authors": [
    "Nicolas Curien"
  ],
  "comment": "To appear in Commun. Math. Phys",
  "categories": [
    "math.PR",
    "math-ph",
    "math.MP"
  ],
  "abstract": "We present a way to study the conformal structure of random planar maps. The main idea is to explore the map along an SLE (Schramm--Loewner evolution) process of parameter $ \\kappa = 6$ and to combine the locality property of the SLE_{6} together with the spatial Markov property of the underlying lattice in order to get a non-trivial geometric information. We follow this path in the case of the conformal structure of random triangulations with a boundary. Under a reasonable assumption called (*) that we have unfortunately not been able to verify, we prove that the limit of uniformized random planar triangulations has a fractal boundary measure of Hausdorff dimension $\\frac{1}{3}$ almost surely. This agrees with the physics KPZ predictions and represents a first step towards a rigorous understanding of the links between random planar maps and the Gaussian free field (GFF).",
  "revisions": [
    {
      "version": "v2",
      "updated": "2014-08-19T10:45:02.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "random planar maps",
      "conformal structure",
      "uniformized random planar triangulations",
      "spatial markov property",
      "non-trivial geometric information"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2013arXiv1308.1807C"
    }
  }
}