{
  "id": "2011.07223",
  "version": "v1",
  "published": "2020-11-14T05:25:34.000Z",
  "updated": "2020-11-14T05:25:34.000Z",
  "title": "Uniform fluctuation and wandering bounds in first passage percolation",
  "authors": [
    "Kenneth S. Alexander"
  ],
  "comment": "86 pages, 12 figures",
  "categories": [
    "math.PR"
  ],
  "abstract": "We consider first passage percolation on certain isotropic random graphs in $\\mathbb{R}^d$. We assume exponential concentration of passage times $T(x,y)$, on some scale $\\sigma_r$ whenever $|y-x|$ is of order $r$, with $\\sigma_r$ \"growning like $r^\\chi$\" for some $0<\\chi<1$. Heuristically this means transverse wandering of geodesics should be at most of order $\\Delta_r = (r\\sigma_r)^{1/2}$. We show that in fact uniform versions of exponential concentration and wandering bounds hold: except with probability exponentially small in $t$, there are no $x,y$ in a natural cylinder of length $r$ and radius $K\\Delta_r$ for which either (i) $|T(x,y) - ET(x,y)|\\geq t\\sigma_r$, or (ii) the geodesic from $x$ to $y$ wanders more than distance $\\sqrt{t}\\Delta_r$ from the cylinder axis. We also establish that for the time constant $\\mu = \\lim_n ET(0,ne_1)/n$, the \"nonrandom error\" $|\\mu|x| - ET(0,x)|$ is at most a constant multiple of $\\sigma(|x|)$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2020-11-14T05:25:34.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "60K35",
      "82B43"
    ],
    "keywords": [
      "first passage percolation",
      "wandering bounds",
      "uniform fluctuation",
      "isotropic random graphs",
      "assume exponential concentration"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 86,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}