{
  "id": "math/0701689",
  "version": "v1",
  "published": "2007-01-24T16:46:36.000Z",
  "updated": "2007-01-24T16:46:36.000Z",
  "title": "Shape curvatures and transversal fluctuations in the first passage percolation model",
  "authors": [
    "Yu Zhang"
  ],
  "comment": "29 pages and 5 figures",
  "categories": [
    "math.PR"
  ],
  "abstract": "We consider the first passage percolation model on the square lattice. In this model, $\\{t(e): e{an edge of}{\\bf Z}^2 \\}$ is an independent identically distributed family with a common distribution $F$. We denote by $T({\\bf 0}, v)$ the passage time from the origin to $v$ for $v\\in {\\bf R}^2$ and $B(t)=\\{v\\in {\\bf R}^d: T({\\bf 0}, v)\\leq t\\}.$ It is well known that if $F(0) < p_c$, there exists a compact shape ${\\bf B}_F\\subset {\\bf R}^2$ such that for all $\\epsilon >0$, $t {\\bf B}_F(1-\\epsilon) \\subset {B(t)} \\subset t{\\bf B}_F(1+\\epsilon)$, eventually with a probability 1. For each shape boundary point $u$, we denote its right- and left-curvature exponents by $\\kappa^+(u)$ and $\\kappa^-(u)$. In addition, for each vector $u$, we denote the transversal fluctuation exponent by $\\xi(u)$. In this paper, we can show that $\\xi(u) \\leq 1-\\max\\{\\kappa^-(u)/2, \\kappa^+(u)/2\\}$ for all shape boundary points $u$. To pursue a curvature on ${\\bf B}_F$, we consider passage times with a special distribution infsupp$(F)=l$ and $F(l)=p > \\vec{p}_c$, where $l$ is a positive number and $\\vec{p}_c$ is a critical point for the oriented percolation model. With this distribution, it is known that there is a flat segment on the shape boundary between angles $0< \\theta_p^- < \\theta_p^+< 90^\\circ$. In this paper, we show that the shape are strictly convex at the directions $\\theta_p^\\pm$. Moreover, we also show that for all $r>0$, $\\xi((r, \\theta^\\pm_p)) = 0.5$ and $\\xi((r, \\theta)) =1$ for all $\\theta_p^- <\\theta< \\theta_p^+$ and $r>0$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2007-01-24T16:46:36.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "60K35"
    ],
    "keywords": [
      "first passage percolation model",
      "transversal fluctuation",
      "shape curvatures",
      "shape boundary point",
      "passage time"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 29,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2007math......1689Z"
    }
  }
}