{
  "id": "1311.5847",
  "version": "v2",
  "published": "2013-11-22T19:23:11.000Z",
  "updated": "2015-02-15T20:43:28.000Z",
  "title": "Liouville Brownian motion at criticality",
  "authors": [
    "Rémi Rhodes",
    "Vincent Vargas"
  ],
  "comment": "52 pages",
  "categories": [
    "math.PR",
    "math-ph",
    "math.MP"
  ],
  "abstract": "In this paper, we construct the Brownian motion of Liouville Quantum Gravity with central charge $c=1$ (more precisely we restrict to the corresponding free field theory). Liouville quantum gravity with $c=1$ corresponds to two-dimensional string theory and is the conjectural scaling limit of large planar maps weighted with a $O(n=2)$ loop model or a $Q=4$-state Potts model embedded in a two dimensional surface in a conformal manner. Following \\cite{GRV1}, we start by constructing the critical LBM from one fixed point $x\\in\\mathbb{R}^2$ (or $x\\in\\S^2$), which amounts to changing the speed of a standard planar Brownian motion depending on the local behaviour of the critical Liouville measure $M'(dx)=-X(x)e^{2X(x)}\\,dx$ (where $X$ is a Gaussian Free Field, say on $\\mathbb{S}^2$). Extending this construction simultaneously to all points in $\\mathbb{R}^2$ requires a fine analysis of the potential properties of the measure $M'$. This allows us to construct a strong Markov process with continuous sample paths living on the support of $M'$, namely a dense set of Hausdorff dimension $0$. We finally construct the associated Liouville semigroup, resolvent, Green function, heat kernel and Dirichlet form. In passing, we extend to quite a general setting the construction of the critical Gaussian multiplicative chaos that was initiated in \\cite{Rnew7,Rnew12} and also establish new capacity estimates for the critical Gaussian multiplicative chaos.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2013-11-22T19:23:11.000Z",
      "abstract": "In this paper, we construct the Brownian motion of Liouville Quantum gravity when the underlying conformal field theory has a $c=1$ central charge. Liouville quantum gravity with $c=1$ corresponds to two-dimensional string theory and is the conjectural scaling limit of large planar maps weighted with a $O(n=2)$ loop model or a $Q=4$-state Potts model embedded in a two dimensional surface in a conformal manner. Following \\cite{GRV1}, we start by constructing the critical LBM from one fixed point $x\\in\\mathbb{R}^2$ (or $x\\in\\mathbb{S}^2$), which amounts to changing the speed of a standard planar Brownian motion depending on the local behaviour of the critical Liouville measure $M'(dx)=-X(x)e^{2X(x)}\\,dx$ (where $X$ is a Gaussian Free Field on $\\mathbb{R}^2$ or $\\mathbb{S}^2$). Extending this construction simultaneously to all points in $\\mathbb{R}^2$ requires a fine analysis of the potential properties of the measure $M'$. This allows us to construct a strong Markov process with continuous sample paths living on the support of $M'$, namely a dense set of Hausdorff dimension $0$. We finally construct the Liouville semigroup, resolvent, Green function, heat kernel and Dirichlet form of (critical) Liouville quantum gravity with a $c=1$ central charge. In passing, we extend to quite a general setting the construction of the critical Gaussian multiplicative chaos that was initiated in \\cite{Rnew7,Rnew12}.",
      "journal": null,
      "doi": null
    },
    {
      "version": "v2",
      "updated": "2015-02-15T20:43:28.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "60J65",
      "81T40"
    ],
    "keywords": [
      "liouville brownian motion",
      "liouville quantum gravity",
      "central charge",
      "criticality",
      "strong markov process"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 52,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "inspire": 1265710,
      "adsabs": "2013arXiv1311.5847R"
    }
  }
}