{
  "id": "1903.09111",
  "version": "v1",
  "published": "2019-03-21T17:03:12.000Z",
  "updated": "2019-03-21T17:03:12.000Z",
  "title": "Liouville quantum gravity with central charge in $(1,25)$: a probabilistic approach",
  "authors": [
    "Ewain Gwynne",
    "Nina Holden",
    "Joshua Pfeffer",
    "Guillaume Remy"
  ],
  "comment": "47 pages, 6 figures",
  "categories": [
    "math.PR",
    "math-ph",
    "math.MP"
  ],
  "abstract": "There is a substantial literature concerning Liouville quantum gravity (LQG) with coupling constant $\\gamma \\in (0,2]$. In this setting, the central charge of the corresponding matter field satisfies $\\mathbf c = 25 - 6(2/\\gamma+\\gamma/2)^2 \\in (-\\infty,1]$. Physics considerations suggest that LQG also makes sense for $\\mathbf c > 1$, but the behavior in this regime is rather mysterious in part because the corresponding value of $\\gamma$ is complex, so analytic continuations of various formulas give non-physical complex answers. We introduce and study a discretization of LQG which makes sense for all $\\mathbf c \\in (-\\infty,25)$. Our discretization consists of a random planar map, defined as the adjacency graph of a tiling of the plane by dyadic squares which all have approximately the same \"LQG size\" with respect to a Gaussian free field. We prove that several formulas for dimension-related quantities are still valid for $\\mathbf c \\in (1,25)$, with the caveat that the dimension is infinite when the formulas give a complex answer. In particular, we prove an extension of the KPZ formula for $\\mathbf c \\in (1,25)$, which gives a finite quantum dimension if and only if the Euclidean dimension is at most $(25-\\mathbf c)/12$. We also show that the graph distance between typical points with respect to our discrete model grows polynomially whereas the cardinality of a graph distance ball of radius $r$ grows faster than any power of $r$ (which suggests that the Hausdorff dimension of LQG for $\\mathbf c\\in(1,25)$ is infinite). We include a substantial list of open problems.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2019-03-21T17:03:12.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "central charge",
      "probabilistic approach",
      "graph distance",
      "literature concerning liouville quantum gravity",
      "complex answer"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 47,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}