{
  "id": "2001.11467",
  "version": "v1",
  "published": "2020-01-30T17:26:54.000Z",
  "updated": "2020-01-30T17:26:54.000Z",
  "title": "Volume of metric balls in Liouville quantum gravity",
  "authors": [
    "Morris Ang",
    "Hugo Falconet",
    "Xin Sun"
  ],
  "comment": "42 pages; 2 figures",
  "categories": [
    "math.PR",
    "math-ph",
    "math.MP"
  ],
  "abstract": "We study the volume of metric balls in Liouville quantum gravity (LQG). For $\\gamma \\in (0,2)$, it has been known since the early work of Kahane (1985) and Molchan (1996) that the LQG volume of Euclidean balls has finite moments exactly for $p \\in (-\\infty, 4/\\gamma^2)$. Here, we prove that the LQG volume of LQG metric balls admits all finite moments. This answers a question of Gwynne and Miller and generalizes a result obtained by Le Gall for the Brownian map, namely, the $\\gamma = \\sqrt{8/3}$ case. We use this moment bound to show that on a compact set the volume of metric balls of size $r$ is given by $r^{d_{\\gamma}+o_r(1)}$, where $d_{\\gamma}$ is the dimension of the LQG metric space. Using similar techniques, we prove analogous results for the first exit time of Liouville Brownian motion from a metric ball. Our result implies that the metric measure space structure of $\\gamma$-LQG determines its conformal structure.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2020-01-30T17:26:54.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "liouville quantum gravity",
      "finite moments",
      "lqg volume",
      "lqg metric balls admits",
      "metric measure space structure"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 42,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}