{
  "id": "0808.1560",
  "version": "v2",
  "published": "2008-08-11T19:39:54.000Z",
  "updated": "2010-12-02T18:36:35.000Z",
  "title": "Liouville Quantum Gravity and KPZ",
  "authors": [
    "Bertrand Duplantier",
    "Scott Sheffield"
  ],
  "comment": "56 pages. Revised version contains more details. To appear in Inventiones",
  "categories": [
    "math.PR",
    "math-ph",
    "math.MP"
  ],
  "abstract": "Consider a bounded planar domain D, an instance h of the Gaussian free field on D (with Dirichlet energy normalized by 1/(2\\pi)), and a constant 0 < gamma < 2. The Liouville quantum gravity measure on D is the weak limit as epsilon tends to 0 of the measures \\epsilon^{\\gamma^2/2} e^{\\gamma h_\\epsilon(z)}dz, where dz is Lebesgue measure on D and h_\\epsilon(z) denotes the mean value of h on the circle of radius epsilon centered at z. Given a random (or deterministic) subset X of D one can define the scaling dimension of X using either Lebesgue measure or this random measure. We derive a general quadratic relation between these two dimensions, which we view as a probabilistic formulation of the KPZ relation from conformal field theory. We also present a boundary analog of KPZ (for subsets of the boundary of D). We discuss the connection between discrete and continuum quantum gravity and provide a framework for understanding Euclidean scaling exponents via quantum gravity.",
  "revisions": [
    {
      "version": "v2",
      "updated": "2010-12-02T18:36:35.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "liouville quantum gravity measure",
      "lebesgue measure",
      "continuum quantum gravity",
      "gaussian free field",
      "general quadratic relation"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 56,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2008arXiv0808.1560D"
    }
  }
}