{
  "id": "1401.0917",
  "version": "v2",
  "published": "2014-01-05T17:52:05.000Z",
  "updated": "2014-11-10T03:17:45.000Z",
  "title": "Subdiffusive concentration in first-passage percolation",
  "authors": [
    "Michael Damron",
    "Jack Hanson",
    "Philippe Sosoe"
  ],
  "comment": "31 pages, the main discrete derivative bound is now simplified, formulated in terms of integrating over uniform variables",
  "categories": [
    "math.PR"
  ],
  "abstract": "We prove exponential concentration in i.i.d. first-passage percolation in $Z^d$ for all $d \\geq 2$ and general edge-weights $(t_e)$. Precisely, under an exponential moment assumption $E e^{\\alpha t_e}< \\infty$ for some $\\alpha>0$) on the edge-weight distribution, we prove the inequality $$ P(|T(0,x)-E T(0,x)| \\geq \\lambda \\sqrt{\\frac{|x|}{log |x|}}) \\leq ce^{-c' \\lambda}, |x|>1 $$ for the point-to-point passage time $T(0,x)$. Under a weaker assumption $E t_e^2(\\log t_e)_+< \\infty$ we show a corresponding inequality for the lower-tail of the distribution of $T(0,x)$. These results extend work of Benaim-Rossignol to general distributions.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2014-01-05T17:52:05.000Z",
      "abstract": "We prove exponential concentration in i.i.d. first-passage percolation in $Z^d$ for all $d \\geq 2$ and general edge-weights $(t_e)$. Precisely, under an exponential moment assumption $E e^{\\alpha t_e}< \\infty$ for some $\\alpha>0$) on the edge-weight distribution, we prove the inequality $$ P\\left(|T(0,x)-E T(0,x)| \\geq \\lambda \\sqrt{\\frac{|x|}{log |x|}}\\right) \\leq ce^{-c' \\lambda},~|x|>1 $$ for the point-to-point passage time $T(0,x)$. Under a weaker assumption $E t_e^2(\\log t_e)_+< \\infty$ we show a corresponding inequality for the lower-tail of the distribution of $T(0,x)$. These results extend work of Benaim-Rossignol to general distributions.",
      "comment": "30 pages",
      "journal": null,
      "doi": null
    },
    {
      "version": "v2",
      "updated": "2014-11-10T03:17:45.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "60K35",
      "60E15"
    ],
    "keywords": [
      "first-passage percolation",
      "subdiffusive concentration",
      "exponential moment assumption",
      "results extend work",
      "point-to-point passage time"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 31,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2014arXiv1401.0917D"
    }
  }
}