{
  "id": "0706.3626",
  "version": "v1",
  "published": "2007-06-25T12:24:50.000Z",
  "updated": "2007-06-25T12:24:50.000Z",
  "title": "A Problem in Last-Passage Percolation",
  "authors": [
    "Harry Kesten",
    "Vladas Sidoravicius"
  ],
  "comment": "1 figure",
  "categories": [
    "math.PR",
    "math-ph",
    "math.MP"
  ],
  "abstract": "Let $\\{X(v), v \\in \\Bbb Z^d \\times \\Bbb Z_+\\}$ be an i.i.d. family of random variables such that $P\\{X(v)= e^b\\}=1-P\\{X(v)= 1\\} = p$ for some $b>0$. We consider paths $\\pi \\subset \\Bbb Z^d \\times \\Bbb Z_+$ starting at the origin and with the last coordinate increasing along the path, and of length $n$. Define for such paths $W(\\pi) = \\text{number of vertices $\\pi_i, 1 \\le i \\le n$, with}X(\\pi_i) = e^b$. Finally let $N_n(\\al) = \\text{number of paths $\\pi$ of length $n$ starting at $\\pi_0 = \\bold 0$ and with $W(\\pi) \\ge \\al n$.}$ We establish several properties of $\\lim_{n \\to \\infty} [N_n]^{1/n}$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2007-06-25T12:24:50.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "60K35",
      "60J15"
    ],
    "keywords": [
      "last-passage percolation",
      "random variables",
      "coordinate"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2007arXiv0706.3626K"
    }
  }
}