{
  "id": "1611.03901",
  "version": "v1",
  "published": "2016-11-11T22:29:11.000Z",
  "updated": "2016-11-11T22:29:11.000Z",
  "title": "Return probability and recurrence for the random walk driven by two-dimensional Gaussian free field",
  "authors": [
    "Marek Biskup",
    "Jian Ding",
    "Subhajit Goswami"
  ],
  "comment": "65 pages, 9 figures",
  "categories": [
    "math.PR"
  ],
  "abstract": "Given any $\\gamma>0$ and for $\\eta=\\{\\eta_v\\}_{v\\in \\mathbb Z^2}$ denoting a sample of the two-dimensional discrete Gaussian free field on $\\mathbb Z^2$ pinned at the origin, we consider the random walk on $\\mathbb Z^2$ among random conductances where the conductance of edge $(u, v)$ is given by $\\mathrm{e}^{\\gamma(\\eta_u + \\eta_v)}$. We show that, for almost every $\\eta$, this random walk is recurrent and that, with probability tending to 1 as $T\\to \\infty$, the return probability at time $2T$ decays as $T^{-1+o(1)}$. In addition, we prove a version of subdiffusive behavior by showing that the expected exit time from a ball of radius $N$ scales as $N^{\\psi(\\gamma)+o(1)}$ with $\\psi(\\gamma)>2$ for all $\\gamma>0$. Our results rely on delicate control of the effective resistance for this random network. In particular, we show that the effective resistance between two vertices at Euclidean distance $N$ behaves as $N^{o(1)}$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2016-11-11T22:29:11.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "60J10",
      "60D05"
    ],
    "keywords": [
      "two-dimensional gaussian free field",
      "random walk driven",
      "return probability",
      "two-dimensional discrete gaussian free field",
      "recurrence"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 65,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}