{
  "id": "1807.01072",
  "version": "v1",
  "published": "2018-07-03T10:37:38.000Z",
  "updated": "2018-07-03T10:37:38.000Z",
  "title": "The fractal dimension of Liouville quantum gravity: universality, monotonicity, and bounds",
  "authors": [
    "Jian Ding",
    "Ewain Gwynne"
  ],
  "comment": "54 pages, 7 figues",
  "categories": [
    "math.PR",
    "math-ph",
    "math.MP"
  ],
  "abstract": "We prove that for each $\\gamma \\in (0,2)$, there is an exponent $d_\\gamma > 2$, the \"fractal dimension of $\\gamma$-Liouville quantum gravity (LQG)\", which describes the ball volume growth exponent for certain random planar maps in the $\\gamma$-LQG universality class, the exponent for the Liouville heat kernel, and exponents for various continuum approximations of $\\gamma$-LQG distances such as Liouville graph distance and Liouville first passage percolation. We also show that $d_\\gamma$ is a continuous, strictly increasing function of $\\gamma$ and prove upper and lower bounds for $d_\\gamma$ which in some cases greatly improve on previously known bounds for the aforementioned exponents. For example, for $\\gamma=\\sqrt 2$ (which corresponds to spanning-tree weighted planar maps) our bounds give $3.4641 \\leq d_{\\sqrt 2} \\leq 3.63299$ and in the limiting case we get $4.77485 \\leq \\lim_{\\gamma\\rightarrow 2^-} d_\\gamma \\leq 4.89898$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2018-07-03T10:37:38.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "liouville quantum gravity",
      "fractal dimension",
      "ball volume growth exponent",
      "liouville first passage percolation",
      "monotonicity"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 54,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}