{
  "id": "1002.0381",
  "version": "v2",
  "published": "2010-02-02T02:19:22.000Z",
  "updated": "2010-06-08T19:25:59.000Z",
  "title": "Fluctuations for the Ginzburg-Landau $\\nabla φ$ Interface Model on a Bounded Domain",
  "authors": [
    "Jason Miller"
  ],
  "comment": "58 pages",
  "categories": [
    "math.PR",
    "math-ph",
    "math.MP"
  ],
  "abstract": "We study the massless field on $D_n = D \\cap \\tfrac{1}{n} \\Z^2$, where $D \\subseteq \\R^2$ is a bounded domain with smooth boundary, with Hamiltonian $\\CH(h) = \\sum_{x \\sim y} \\CV(h(x) - h(y))$. The interaction $\\CV$ is assumed to be symmetric and uniformly convex. This is a general model for a $(2+1)$-dimensional effective interface where $h$ represents the height. We take our boundary conditions to be a continuous perturbation of a macroscopic tilt: $h(x) = n x \\cdot u + f(x)$ for $x \\in \\partial D_n$, $u \\in \\R^2$, and $f \\colon \\R^2 \\to \\R$ continuous. We prove that the fluctuations of linear functionals of $h(x)$ about the tilt converge in the limit to a Gaussian free field on $D$, the standard Gaussian with respect to the weighted Dirichlet inner product $(f,g)_\\nabla^\\beta = \\int_D \\sum_i \\beta_i \\partial_i f_i \\partial_i g_i$ for some explicit $\\beta = \\beta(u)$. In a subsequent article, we will employ the tools developed here to resolve a conjecture of Sheffield that the zero contour lines of $h$ are asymptotically described by $SLE(4)$, a conformally invariant random curve.",
  "revisions": [
    {
      "version": "v2",
      "updated": "2010-06-08T19:25:59.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "60F05",
      "60G60",
      "60J27"
    ],
    "keywords": [
      "bounded domain",
      "interface model",
      "fluctuations",
      "ginzburg-landau",
      "conformally invariant random curve"
    ],
    "tags": [
      "journal article"
    ],
    "publication": {
      "doi": "10.1007/s00220-011-1315-9",
      "journal": "Communications in Mathematical Physics",
      "year": 2011,
      "month": "Dec",
      "volume": 308,
      "number": 3,
      "pages": 591
    },
    "note": {
      "typesetting": "TeX",
      "pages": 58,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2011CMaPh.308..591M"
    }
  }
}