{
  "id": "1406.1108",
  "version": "v1",
  "published": "2014-06-04T16:50:28.000Z",
  "updated": "2014-06-04T16:50:28.000Z",
  "title": "Variational formula for the time-constant of first-passage percolation",
  "authors": [
    "Arjun Krishnan"
  ],
  "comment": "112 pages, double spaced, 2 figures. PhD Thesis, Courant Institute, New York University",
  "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. We then construct an explicit iteration that produces the minimizer of the variational formula (under a symmetry assumption), thereby computing the time-constant. The variational formula may also be seen as a duality principle, and we discuss some aspects of this duality.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2014-06-04T16:50:28.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "60K35",
      "82B43"
    ],
    "keywords": [
      "first-passage percolation",
      "time-constant",
      "exact variational formula",
      "scaled first-passage time",
      "stationary-ergodic weights"
    ],
    "tags": [
      "dissertation"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 112,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2014arXiv1406.1108K"
    }
  }
}