{
  "id": "1008.2447",
  "version": "v1",
  "published": "2010-08-14T13:54:21.000Z",
  "updated": "2010-08-14T13:54:21.000Z",
  "title": "A contour line of the continuum Gaussian free field",
  "authors": [
    "Oded Schramm",
    "Scott Sheffield"
  ],
  "comment": "44 pages, 1 figure",
  "categories": [
    "math.PR",
    "math-ph",
    "math.MP"
  ],
  "abstract": "Consider an instance $h$ of the Gaussian free field on a simply connected planar domain with boundary conditions $-\\lambda$ on one boundary arc and $\\lambda$ on the complementary arc, where $\\lambda$ is the special constant $\\sqrt{\\pi/8}$. We argue that even though $h$ is defined only as a random distribution, and not as a function, it has a well-defined zero contour line connecting the endpoints of these arcs, whose law is SLE(4). We construct this contour line in two ways: as the limit of the chordal zero contour lines of the projections of $h$ onto certain spaces of piecewise linear functions, and as the only path-valued function on the space of distributions with a natural Markov property.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2010-08-14T13:54:21.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "continuum gaussian free field",
      "zero contour line connecting",
      "chordal zero contour lines",
      "well-defined zero contour line",
      "natural markov property"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 44,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2010arXiv1008.2447S"
    }
  }
}