{
  "id": "math/0501095",
  "version": "v1",
  "published": "2005-01-06T22:38:03.000Z",
  "updated": "2005-01-06T22:38:03.000Z",
  "title": "The divergence of fluctuations for the shape on first passage percolation",
  "authors": [
    "Yu Zhang"
  ],
  "categories": [
    "math.PR"
  ],
  "abstract": "Consider the first passage percolation model on ${\\bf Z}^d$ for $d\\geq 2$. In this model we assign independently to each edge the value zero with probability $p$ and the value one with probability $1-p$. We denote by $T({\\bf 0}, v)$ the passage time from the origin to $v$ for $v\\in {\\bf R}^d$ and $$B(t)=\\{v\\in {\\bf R}^d: T({\\bf 0}, v)\\leq t\\}{and} G(t)=\\{v\\in {\\bf R}^d: ET({\\bf 0}, v)\\leq t\\}.$$ It is well known that if $p < p_c$, there exists a compact shape $B_d\\subset {\\bf R}^d$ such that for all $\\epsilon >0$ $$t B_d(1-\\epsilon) \\subset {B(t)} \\subset tB_d(1+\\epsilon){and} G(t)(1-{\\epsilon}) \\subset {B(t)} \\subset G(t)(1+{\\epsilon}) {eventually w.p.1.}$$ We denote the fluctuations of $B(t)$ from $tB_d$ and $G(t)$ by &&F(B(t), tB_d)=\\inf \\{l:tB_d(1-{l\\over t})\\subset B(t)\\subset tB_d(1+{l\\over t})\\} && F(B(t), G(t))=\\inf\\{l:G(t)(1-{l\\over t})\\subset B(t)\\subset G(t)(1+{l\\over t})\\}. The means of the fluctuations $E[F(B(t), tB_d]$ and $E[F(B(t), G(t))]$ have been conjectured ranging from divergence to non-divergence for large $d\\geq 2$ by physicists. In this paper, we show that for all $d\\geq 2$ with a high probability, the fluctuations $F(B(t), G(t))$ and $F(B(t), tB_d)$ diverge with a rate of at least $C \\log t$ for some constant $C$. The proof of this argument depends on the linearity between the number of pivotal edges of all minimizing paths and the paths themselves. This linearity is also independently interesting.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2005-01-06T22:38:03.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "60K35"
    ],
    "keywords": [
      "fluctuations",
      "divergence",
      "first passage percolation model",
      "pivotal edges",
      "value zero"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2005math......1095Z"
    }
  }
}