{
  "id": "1709.09613",
  "version": "v1",
  "published": "2017-09-27T16:40:22.000Z",
  "updated": "2017-09-27T16:40:22.000Z",
  "title": "The size of the boundary in first-passage percolation",
  "authors": [
    "Michael Damron",
    "Jack Hanson",
    "Wai-Kit Lam"
  ],
  "categories": [
    "math.PR"
  ],
  "abstract": "First-passage percolation is a random growth model defined using i.i.d. edge-weights $(t_e)$ on the nearest-neighbor edges of $\\mathbb{Z}^d$. An initial infection occupies the origin and spreads along the edges, taking time $t_e$ to cross the edge $e$. In this paper, we study the size of the boundary of the infected (\"wet\") region at time $t$, $B(t)$. It is known that $B(t)$ grows linearly, so its boundary $\\partial B(t)$ has size between $ct^{d-1}$ and $Ct^d$. Under a weak moment condition on the weights, we show that for most times, $\\partial B(t)$ has size of order $t^{d-1}$ (smooth). On the other hand, for heavy-tailed distributions, $B(t)$ contains many small holes, and consequently we show that $\\partial B(t)$ has size of order $t^{d-1+\\alpha}$ for some $\\alpha>0$ depending on the distribution. In all cases, we show that the exterior boundary of $B(t)$ (edges touching the unbounded component of the complement of $B(t)$) is smooth for most times. Under the unproven assumption of uniformly positive curvature on the limit shape for $B(t)$, we show the inequality $\\#\\partial B(t) \\leq (\\log t)^C t^{d-1}$ for all large $t.$",
  "revisions": [
    {
      "version": "v1",
      "updated": "2017-09-27T16:40:22.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "60K35",
      "82B43"
    ],
    "keywords": [
      "first-passage percolation",
      "random growth model",
      "initial infection occupies",
      "weak moment condition",
      "exterior boundary"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}