{
  "id": "1705.03573",
  "version": "v1",
  "published": "2017-05-10T00:25:24.000Z",
  "updated": "2017-05-10T00:25:24.000Z",
  "title": "Schnyder woods, SLE(16), and Liouville quantum gravity",
  "authors": [
    "Yiting Li",
    "Xin Sun",
    "Samuel S. Watson"
  ],
  "comment": "51 pages, 23 figures",
  "categories": [
    "math.PR"
  ],
  "abstract": "In 1990, Schnyder used a 3-spanning-tree decomposition of a simple triangulation, now known as the Schnyder wood, to give a fundamental grid-embedding algorithm for planar maps. We show that a uniformly sampled Schnyder-wood-decorated triangulation with $n$ vertices converges as $n\\to\\infty$ to the Liouville quantum gravity with parameter $1$, decorated with a triple of SLE$_{16}$'s of angle difference $2\\pi/3$ in the imaginary geometry sense. Our convergence result provides a description of the continuum limit of Schnyder's embedding algorithm via Liouville quantum gravity and imaginary geometry.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2017-05-10T00:25:24.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "60C05",
      "68R10"
    ],
    "keywords": [
      "liouville quantum gravity",
      "schnyder wood",
      "imaginary geometry sense",
      "planar maps",
      "schnyders embedding algorithm"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 51,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}