{
  "id": "1310.1247",
  "version": "v2",
  "published": "2013-10-04T12:57:32.000Z",
  "updated": "2014-03-15T16:07:15.000Z",
  "title": "Law of large numbers for critical first-passage percolation on the triangular lattice",
  "authors": [
    "Chang-Long Yao"
  ],
  "comment": "14 pages, 2 figures",
  "journal": "Electronic Communications in Probability 19 (2014) 1--14",
  "categories": [
    "math.PR",
    "math-ph",
    "math.MP"
  ],
  "abstract": "We study the site version of (independent) first-passage percolation on the triangular lattice $\\mathbb{T}$. Denote the passage time of the site $v$ in $\\mathbb{T}$ by $t(v)$, and assume that $P(t(v)=0)=P(t(v)=1)=1/2$. Denote by $a_{0,n}$ the passage time from $\\textbf{0}$ to $(n,0)$, and by $b_{0,n}$ the passage time from $\\textbf{0}$ to the halfplane $\\{(x,y):x\\geq n\\}$. We prove that there exists a constant $0<\\mu<\\infty$ such that as $n\\rightarrow\\infty$, $a_{0,n}/\\log n\\rightarrow \\mu$ in probability and $b_{0,n}/\\log n\\rightarrow \\mu/2$ almost surely. This result confirms a prediction of Kesten and Zhang (Probab. Theory Relat. Fields \\textbf{107}: 137--160, 1997). The proof relies on the existence of the full scaling limit of critical site percolation on $\\mathbb{T}$, established by Camia and Newman.",
  "revisions": [
    {
      "version": "v2",
      "updated": "2014-03-15T16:07:15.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "critical first-passage percolation",
      "triangular lattice",
      "large numbers",
      "passage time",
      "site version"
    ],
    "tags": [
      "journal article"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 14,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2013arXiv1310.1247Y"
    }
  }
}