{
  "id": "1903.09561",
  "version": "v1",
  "published": "2019-03-22T15:37:28.000Z",
  "updated": "2019-03-22T15:37:28.000Z",
  "title": "Bounds for distances and geodesic dimension in Liouville first passage percolation",
  "authors": [
    "Ewain Gwynne",
    "Josh Pfeffer"
  ],
  "comment": "10 pages, 2 figures",
  "categories": [
    "math.PR",
    "math-ph",
    "math.MP"
  ],
  "abstract": "For $\\xi \\geq 0$, Liouville first passage percolation (LFPP) is the random metric on $\\varepsilon \\mathbb Z^2$ obtained by weighting each vertex by $\\varepsilon e^{\\xi h_\\varepsilon(z)}$, where $h_\\varepsilon(z)$ is the average of the Gaussian free field $h$ over the circle $\\partial B_\\varepsilon(z)$. Ding and Gwynne (2018) showed that for $\\gamma \\in (0,2)$, LFPP with parameter $\\xi = \\gamma/d_\\gamma$ is related to $\\gamma$-Liouville quantum gravity (LQG), where $d_\\gamma$ is the $\\gamma$-LQG dimension exponent (which is expected to be the Hausdorff dimension of the conjectural LQG metric). For $\\xi > 2/d_2$, LFPP is instead expected to be related to LQG with central charge greater than 1. We prove several estimates for LFPP distances for general $\\xi\\geq 0$. For $\\xi\\leq 2/d_2$, this leads to new bounds for $d_\\gamma$ which improve on the best previously known upper (resp.\\ lower) bounds for $d_\\gamma$ in the case when $\\gamma > \\sqrt{8/3}$ (resp.\\ $\\gamma \\in (0.4981, \\sqrt{8/3})$). These bounds are consistent with the Watabiki (1993) prediction for $d_\\gamma$. However, for $\\xi > 1/\\sqrt 3$ (or equivalently for LQG with central charge larger than 17) our bounds are inconsistent with the analytic continuation of Watabiki's prediction to the $\\xi >2/d_2$ regime. We also obtain an upper bound for the Euclidean dimension of LFPP geodesics.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2019-03-22T15:37:28.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "liouville first passage percolation",
      "geodesic dimension",
      "liouville quantum gravity",
      "central charge larger",
      "lqg dimension exponent"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 10,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}