{
  "id": "1606.01214",
  "version": "v1",
  "published": "2016-06-03T18:30:36.000Z",
  "updated": "2016-06-03T18:30:36.000Z",
  "title": "A distance exponent for Liouville quantum gravity",
  "authors": [
    "Ewain Gwynne",
    "Nina Holden",
    "Xin Sun"
  ],
  "comment": "69 pages, 13 figures",
  "categories": [
    "math.PR",
    "math-ph",
    "math.MP"
  ],
  "abstract": "Let $\\gamma \\in (0,2)$ and let $h$ be a random distribution on $\\mathbb C$ which parametrizes a $\\gamma$-Liouville quantum gravity (LQG) cone. Also let $\\kappa = 16/\\gamma^2 >4$ and let $\\eta$ be a whole-plane space-filling SLE$_\\kappa$ curve independent from $h$ and parametrized by $\\gamma$-quantum mass with respect to $h$. We study a family $\\{\\mathcal G^\\epsilon\\}_{\\epsilon>0}$ of planar maps associated with $(h, \\eta)$, which we call the LQG structure graphs, and which we conjecture converges in the scaling limit in the Gromov-Hausdorff topology to a metric on the $\\gamma$-LQG cone. In particular, for $\\mathcal G^\\epsilon$ is the graph whose vertices are segments of the form $\\eta([(k-1)\\epsilon , k\\epsilon])$ for $k\\in\\mathbb Z$, with two such segments connected by an edge if and only if they share a non-trivial boundary arc. Due to the peanosphere description of SLE-decorated LQG due to Duplantier, Miller, and Sheffield (2014), the graph $\\mathcal G^\\epsilon$ can equivalently be expressed as an explicit functional of a certain correlated two-dimensional Brownian motion, so can be studied without any reference to SLE or LQG. We prove that there is an exponent $\\chi > 0$ for which the expected graph distance between generic points in the subgraph of $\\mathcal G^\\epsilon$ corresponding to the segment $\\eta([0,1])$ is of order $\\epsilon^{-\\chi + o_\\epsilon(1)}$, and this distance is extremely unlikely to be larger than $\\epsilon^{-\\chi + o_\\epsilon(1)}$. In the special case when $\\gamma = \\sqrt 2$, we show that the diameter of this subgraph of $\\mathcal G^\\epsilon$ is of order $\\epsilon^{-\\chi + o_\\epsilon(1)}$ with high probability. We also prove non-trivial upper and lower bounds for the cardinality of a graph-distance ball of radius $n$ in $\\mathcal G^\\epsilon$ which are consistent with the prediction of Watabiki (1993) for the Hausdorff dimension of LQG.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2016-06-03T18:30:36.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "liouville quantum gravity",
      "distance exponent",
      "non-trivial boundary arc",
      "correlated two-dimensional brownian motion",
      "lqg structure graphs"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 69,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}