{
  "id": "1311.0316",
  "version": "v2",
  "published": "2013-11-01T22:05:23.000Z",
  "updated": "2014-11-02T23:33:45.000Z",
  "title": "Variational formula for the time-constant of first-passage percolation",
  "authors": [
    "Arjun Krishnan"
  ],
  "comment": "40 pages, 2 figures. I have chosen not to write a part 2. Instead, I've compiled the application into one longer paper",
  "categories": [
    "math.PR"
  ],
  "abstract": "We consider first-passage percolation with positive, stationary-ergodic weights on the square lattice $\\mathbb{Z}^d$. Let $T(x)$ be the first-passage time from the origin to a point $x$ in $\\mathbb{Z}^d$. The convergence of the scaled first-passage time $T([nx])/n$ to the time-constant as $n$ tends to infinity can be viewed as a problem of homogenization for a discrete Hamilton-Jacobi-Bellman (HJB) equation. By borrowing several tools from the continuum theory of stochastic homogenization for HJB equations, we derive an exact variational formula for the time-constant. As an application, we construct an explicit iteration that produces a minimizer of the variational formula (under a symmetry assumption), thereby computing the time-constant. In certain situations, the iteration produces correctors.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2013-11-01T22:05:23.000Z",
      "title": "Variational formula for the time-constant of first-passage percolation I: Homogenization",
      "abstract": "We consider first-passage percolation with positive, stationary-ergodic weights on the square lattice $\\mathbb{Z}^d$. Let $T(x)$ be the first-passage time from the origin to $x \\in \\mathbb{Z}^d$. The convergence of $T([nx])/n$ to the time constant $\\mu(x)$ as $n \\to \\infty$ can be viewed as a problem of homogenization for a discrete Hamilton-Jacobi-Bellman (HJB) equation. By borrowing several tools from the continuum theory of stochastic homogenization for HJB equations, we derive an exact variational formula for $\\mu(x)$. Several applications of the formula are shown in part II of this paper.",
      "comment": "28 pages, 1 figure",
      "journal": null,
      "doi": null
    },
    {
      "version": "v2",
      "updated": "2014-11-02T23:33:45.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "60K35",
      "82B43",
      "35F21"
    ],
    "keywords": [
      "first-passage percolation",
      "time-constant",
      "exact variational formula",
      "first-passage time",
      "time constant"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 40,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2013arXiv1311.0316K"
    }
  }
}