{
  "id": "math/0506241",
  "version": "v2",
  "published": "2005-06-13T13:50:19.000Z",
  "updated": "2006-05-24T12:51:55.000Z",
  "title": "Singularity points for first passage percolation",
  "authors": [
    "J. E. Yukich",
    "Yu Zhang"
  ],
  "comment": "Published at http://dx.doi.org/10.1214/009117905000000819 in the Annals of Probability (http://www.imstat.org/aop/) by the Institute of Mathematical Statistics (http://www.imstat.org)",
  "journal": "Annals of Probability 2006, Vol. 34, No. 2, 577-592",
  "doi": "10.1214/009117905000000819",
  "categories": [
    "math.PR"
  ],
  "abstract": "Let $0<a<b<\\infty$ be fixed scalars. Assign independently to each edge in the lattice $\\mathbb{Z}^2$ the value $a$ with probability $p$ or the value $b$ with probability $1-p$. For all $u,v\\in\\mathbb{Z}^2$, let $T(u,v)$ denote the first passage time between $u$ and $v$. We show that there are points $x\\in\\mathbb{R}^2$ such that the ``time constant'' in the direction of $x$, namely, $\\lim_{n\\to\\infty}n^{-1}\\mathbf{E}_p[T(\\mathbf{0},nx)],$ is not a three times differentiable function of $p$.",
  "revisions": [
    {
      "version": "v2",
      "updated": "2006-05-24T12:51:55.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "60K35"
    ],
    "keywords": [
      "first passage percolation",
      "singularity points",
      "first passage time",
      "probability",
      "time constant"
    ],
    "tags": [
      "journal article"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2005math......6241Y"
    }
  }
}