{
  "id": "1512.01223",
  "version": "v1",
  "published": "2015-12-03T20:46:59.000Z",
  "updated": "2015-12-03T20:46:59.000Z",
  "title": "An almost sure KPZ relation for SLE and Brownian motion",
  "authors": [
    "Ewain Gwynne",
    "Nina Holden",
    "Jason Miller"
  ],
  "comment": "47 page, 12 figures",
  "categories": [
    "math.PR",
    "math-ph",
    "math.CV",
    "math.MP"
  ],
  "abstract": "The peanosphere construction of Duplantier, Miller, and Sheffield provides a means of representing a $\\gamma$-Liouville quantum gravity (LQG) surface, $\\gamma \\in (0,2)$, decorated with a space-filling form of Schramm's SLE$_\\kappa$, $\\kappa = 16/\\gamma^2 \\in (4,\\infty)$, $\\eta$ as a gluing of a pair of trees which are encoded by a correlated two-dimensional Brownian motion $Z$. We prove a KPZ-type formula which relates the Hausdorff dimension of any Borel subset $A$ of the range of $\\eta$ which can be defined as a function of $\\eta$ (modulo time parameterization) to the Hausdorff dimension of the corresponding time set $\\eta^{-1}(A)$. This result serves to reduce the problem of computing the Hausdorff dimension of any set associated with an SLE, CLE, or related processes in the interior of a domain to the problem of computing the Hausdorff dimension of a certain set associated with a Brownian motion. For many natural examples, the associated Brownian motion set is well-known. As corollaries, we obtain new proofs of the Hausdorff dimensions of the SLE$_\\kappa$ curve for $\\kappa \\not=4$; the double points and cut points of SLE$_\\kappa$ for $\\kappa >4$; and the intersection of two flow lines of a Gaussian free field. We also obtain the Hausdorff dimension of the set of $m$-tuple points of space-filling SLE$_\\kappa$ for $\\kappa>4$ and $m \\geq 3$ by computing the Hausdorff dimension of the so-called $(m-2)$-tuple $\\pi/2$-cone times of a correlated planar Brownian motion.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2015-12-03T20:46:59.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "hausdorff dimension",
      "sure kpz relation",
      "associated brownian motion set",
      "correlated planar brownian motion",
      "modulo time parameterization"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 47,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}