{
  "id": "1606.07055",
  "version": "v1",
  "published": "2016-06-22T19:53:17.000Z",
  "updated": "2016-06-22T19:53:17.000Z",
  "title": "Dimension of the SLE light cone, the SLE fan, and SLE$_κ(ρ)$ for $κ\\in (0,4)$ and $ρ\\in [\\tfracκ{2}-4,-2)$",
  "authors": [
    "Jason Miller"
  ],
  "comment": "43 pages, 15 figures",
  "categories": [
    "math.PR",
    "math-ph",
    "math.CV",
    "math.MP"
  ],
  "abstract": "Suppose that $h$ is a Gaussian free field (GFF) on a planar domain. Fix $\\kappa \\in (0,4)$. The SLE$_\\kappa$ light cone ${\\mathbf L}(\\theta)$ of $h$ with opening angle $\\theta \\in [0,\\pi]$ is the set of points reachable from a given boundary point by angle-varying flow lines of the (formal) vector field $e^{i h/\\chi}$, $\\chi = \\tfrac{2}{\\sqrt{\\kappa}} - \\tfrac{\\sqrt{\\kappa}}{2}$, with angles in $[-\\tfrac{\\theta}{2},\\tfrac{\\theta}{2}]$. We derive the Hausdorff dimension of ${\\mathbf L}(\\theta)$. If $\\theta =0$ then ${\\mathbf L}(\\theta)$ is an ordinary SLE$_{\\kappa}$ curve (with $\\kappa < 4$); if $\\theta = \\pi$ then ${\\mathbf L}(\\theta)$ is the range of an SLE$_{\\kappa'}$ curve ($\\kappa' = 16/\\kappa > 4$). In these extremes, this leads to a new proof of the Hausdorff dimension formula for SLE. We also consider SLE$_\\kappa(\\rho)$ processes, which were originally only defined for $\\rho > -2$, but which can also be defined for $\\rho \\leq -2$ using L\\'evy compensation. The range of an SLE$_\\kappa(\\rho)$ is qualitatively different when $\\rho \\leq -2$. In particular, these curves are self-intersecting for $\\kappa < 4$ and double points are dense, while ordinary SLE$_\\kappa$ is simple. It was previously shown (Miller-Sheffield, 2016) that certain SLE$_\\kappa(\\rho)$ curves agree in law with certain light cones. Combining this with other known results, we obtain a general formula for the Hausdorff dimension of SLE$_\\kappa(\\rho)$ for all values of $\\rho$. Finally, we show that the Hausdorff dimension of the so-called SLE$_\\kappa$ fan is the same as that of ordinary SLE$_\\kappa$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2016-06-22T19:53:17.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "sle light cone",
      "sle fan",
      "ordinary sle",
      "hausdorff dimension formula",
      "gaussian free field"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 43,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}