{
  "id": "1602.00065",
  "version": "v1",
  "published": "2016-01-30T02:54:25.000Z",
  "updated": "2016-01-30T02:54:25.000Z",
  "title": "Limit theorems for critical first-passage percolation on the triangular lattice",
  "authors": [
    "Chang-Long Yao"
  ],
  "comment": "17 pages, 1 figure",
  "categories": [
    "math.PR"
  ],
  "abstract": "Consider (independent) first-passage percolation on the sites of 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 $b_{0,n}$ the passage time from 0 to the halfplane $\\{v\\in\\mathbb{T}:\\mbox{Re}(v)\\geq n\\}$, and by $T(0,nu)$ the passage time from 0 to the nearest site to $nu$, where $|u|=1$. We prove that as $n\\rightarrow\\infty$, $b_{0,n}/\\log n\\rightarrow 1/(2\\sqrt{3}\\pi)$ a.s., $E[b_{0,n}]/\\log n\\rightarrow 1/(2\\sqrt{3}\\pi)$ and Var$[b_{0,n}]/\\log n\\rightarrow 2/(3\\sqrt{3}\\pi)-1/(2\\pi^2)$; $T(0,nu)/\\log n\\rightarrow 1/(\\sqrt{3}\\pi)$ in probability but not a.s., $E[T(0,nu)]/\\log n\\rightarrow 1/(\\sqrt{3}\\pi)$ and Var$[T(0,nu)]/\\log n\\rightarrow 4/(3\\sqrt{3}\\pi)-1/\\pi^2$. This answers a question of Kesten and Zhang (1997) and improves our previous work (2014). From this result, we derive explicit form of the central limit theorem for $b_{0,n}$ and $T(0,nu)$. A key ingredient for the proof is the moment generating function of the conformal radii for conformal loop ensemble CLE$_6$, given by Schramm, Sheffield and Wilson (2009).",
  "revisions": [
    {
      "version": "v1",
      "updated": "2016-01-30T02:54:25.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "60K35",
      "82B43"
    ],
    "keywords": [
      "critical first-passage percolation",
      "triangular lattice",
      "passage time",
      "conformal loop ensemble cle",
      "central limit theorem"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 17,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2016arXiv160200065Y"
    }
  }
}