{
  "id": "1603.01194",
  "version": "v1",
  "published": "2016-03-03T17:35:29.000Z",
  "updated": "2016-03-03T17:35:29.000Z",
  "title": "Joint scaling limit of a bipolar-oriented triangulation and its dual in the peanosphere sense",
  "authors": [
    "Ewain Gwynne",
    "Nina Holden",
    "Xin Sun"
  ],
  "comment": "55 pages, 9 figures",
  "categories": [
    "math.PR",
    "math-ph",
    "math.CV",
    "math.MP"
  ],
  "abstract": "Kenyon, Miller, Sheffield, and Wilson (2015) showed how to encode a random bipolar-oriented planar map by means of a random walk with a certain step size distribution. Using this encoding together with the mating-of-trees construction of Liouville quantum gravity due to Duplantier, Miller, and Sheffield (2014), they proved that random bipolar-oriented planar maps converge in the scaling limit to a $\\sqrt{4/3}$-Liouville quantum gravity (LQG) surface decorated by an independent SLE$_{12}$ in the peanosphere sense, meaning that the height functions of a particular pair of trees on the maps converge in the scaling limit to the correlated planar Brownian motion which encodes the SLE-decorated LQG surface. We improve this convergence result by proving that the pair of height functions for an infinite-volume random bipolar-oriented triangulation and the pair of height functions for its dual map converge jointly in law in the scaling limit to the two planar Brownian motions which encode the same $\\sqrt{4/3}$-LQG surface decorated by both an SLE$_{12}$ curve and the \"dual\" SLE$_{12}$ curve which travels in a direction perpendicular (in the sense of imaginary geometry) to the original curve. This confirms a conjecture of Kenyon, Miller, Sheffield, and Wilson (2015).",
  "revisions": [
    {
      "version": "v1",
      "updated": "2016-03-03T17:35:29.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "joint scaling limit",
      "peanosphere sense",
      "bipolar-oriented triangulation",
      "bipolar-oriented planar maps converge",
      "random bipolar-oriented planar map"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 55,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2016arXiv160301194G"
    }
  }
}