{
  "id": "0906.3804",
  "version": "v1",
  "published": "2009-06-22T16:14:28.000Z",
  "updated": "2009-06-22T16:14:28.000Z",
  "title": "The natural parametrization for the Schramm-Loewner evolution",
  "authors": [
    "Gregory F. Lawler",
    "Scott Sheffield"
  ],
  "categories": [
    "math.PR"
  ],
  "abstract": "The Schramm-Loewner evolution (SLE_\\kappa) is a candidate for the scaling limit of random curves arising in two-dimensional critical phenomena. When \\kappa < 8, an instance of SLE_\\kappa is a random planar curve with almost sure Hausdorff dimension d = 1 + \\kappa/8 < 2. This curve is conventionally parametrized by its half plane capacity, rather than by any measure of its d-dimensional volume. For \\kappa < 8, we use a Doob-Meyer decomposition to construct the unique (under mild assumptions) Markovian parametrization of SLE_\\kappa that transforms like a d-dimensional volume measure under conformal maps. We prove that this parametrization is non-trivial (i.e., the curve is not entirely traversed in zero time) for \\kappa < 4(7 - \\sqrt{33}) = 5.021 ....",
  "revisions": [
    {
      "version": "v1",
      "updated": "2009-06-22T16:14:28.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "82B27"
    ],
    "keywords": [
      "schramm-loewner evolution",
      "natural parametrization",
      "d-dimensional volume measure",
      "sure hausdorff dimension",
      "half plane capacity"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2009arXiv0906.3804L"
    }
  }
}