{
  "id": "1010.1356",
  "version": "v1",
  "published": "2010-10-07T07:21:38.000Z",
  "updated": "2010-10-07T07:21:38.000Z",
  "title": "Universality for SLE(4)",
  "authors": [
    "Jason Miller"
  ],
  "comment": "58 pages",
  "categories": [
    "math.PR",
    "math-ph",
    "math.MP"
  ],
  "abstract": "We resolve a conjecture of Sheffield that $\\SLE(4)$, a conformally invariant random curve, is the universal limit of the chordal zero-height contours of random surfaces with isotropic, uniformly convex potentials. Specifically, we study the \\emph{Ginzburg-Landau $\\nabla \\phi$ interface model} or \\emph{anharmonic crystal} on $D_n = D \\cap \\tfrac{1}{n} \\Z^2$ for $D \\subseteq \\C$ a bounded, simply connected Jordan domain with smooth boundary. This is the massless field with Hamiltonian $\\CH(h) = \\sum_{x \\sim y} \\CV(h(x) - h(y))$ with $\\CV$ symmetric and uniformly convex and $h(x) = \\phi(x)$ for $x \\in \\partial D_n$, $\\phi \\colon \\partial D_n \\to \\R$ a given function. We show that the macroscopic chordal contours of $h$ are asymptotically described by $\\SLE(4)$ for appropriately chosen $\\phi$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2010-10-07T07:21:38.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "universality",
      "macroscopic chordal contours",
      "conformally invariant random curve",
      "chordal zero-height contours",
      "uniformly convex"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 58,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2010arXiv1010.1356M"
    }
  }
}