{
  "id": "0902.3842",
  "version": "v3",
  "published": "2009-02-23T03:34:33.000Z",
  "updated": "2010-10-02T06:40:59.000Z",
  "title": "Thick points of the Gaussian free field",
  "authors": [
    "Xiaoyu Hu",
    "Jason Miller",
    "Yuval Peres"
  ],
  "comment": "Published in at http://dx.doi.org/10.1214/09-AOP498 the Annals of Probability (http://www.imstat.org/aop/) by the Institute of Mathematical Statistics (http://www.imstat.org)",
  "journal": "Annals of Probability 2010, Vol. 38, No. 2, 896-926",
  "doi": "10.1214/09-AOP498",
  "categories": [
    "math.PR",
    "math-ph",
    "math.MP"
  ],
  "abstract": "Let $U\\subseteq\\mathbf{C}$ be a bounded domain with smooth boundary and let $F$ be an instance of the continuum Gaussian free field on $U$ with respect to the Dirichlet inner product $\\int_U\\nabla f(x)\\cdot \\nabla g(x)\\,dx$. The set $T(a;U)$ of $a$-thick points of $F$ consists of those $z\\in U$ such that the average of $F$ on a disk of radius $r$ centered at $z$ has growth $\\sqrt{a/\\pi}\\log\\frac{1}{r}$ as $r\\to 0$. We show that for each $0\\leq a\\leq2$ the Hausdorff dimension of $T(a;U)$ is almost surely $2-a$, that $\\nu_{2-a}(T(a;U))=\\infty$ when $0<a\\leq2$ and $\\nu_2(T(0;U))=\\nu_2(U)$ almost surely, where $\\nu_{\\alpha}$ is the Hausdorff-$\\alpha$ measure, and that $T(a;U)$ is almost surely empty when $a>2$. Furthermore, we prove that $T(a;U)$ is invariant under conformal transformations in an appropriate sense. The notion of a thick point is connected to the Liouville quantum gravity measure with parameter $\\gamma$ given formally by $\\Gamma(dz)=e^{\\sqrt{2\\pi}\\gamma F(z)}\\,dz$ considered by Duplantier and Sheffield.",
  "revisions": [
    {
      "version": "v3",
      "updated": "2010-10-02T06:40:59.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "thick point",
      "continuum gaussian free field",
      "liouville quantum gravity measure",
      "dirichlet inner product",
      "conformal transformations"
    ],
    "tags": [
      "journal article"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2009arXiv0902.3842H"
    }
  }
}