{
  "id": "0801.0967",
  "version": "v2",
  "published": "2008-01-07T13:28:08.000Z",
  "updated": "2009-07-03T13:36:56.000Z",
  "title": "Lower large deviations and laws of large numbers for maximal flows through a box in first passage percolation",
  "authors": [
    "Raphaël Rossignol",
    "Marie Théret"
  ],
  "comment": "39 pages, 4 figures; improvement of the moment conditions and introduction of new results in the revised version",
  "categories": [
    "math.PR"
  ],
  "abstract": "We consider the standard first passage percolation model in $\\mathbb{Z}^d$ for $d\\geq 2$. We are interested in two quantities, the maximal flow $\\tau$ between the lower half and the upper half of the box, and the maximal flow $\\phi$ between the top and the bottom of the box. A standard subadditive argument yields the law of large numbers for $\\tau$ in rational directions. Kesten and Zhang have proved the law of large numbers for $\\tau$ and $\\phi$ when the sides of the box are parallel to the coordinate hyperplanes: the two variables grow linearly with the surface $s$ of the basis of the box, with the same deterministic speed. We study the probabilities that the rescaled variables $\\tau /s$ and $\\phi /s$ are abnormally small. For $\\tau$, the box can have any orientation, whereas for $\\phi$, we require either that the box is sufficiently flat, or that its sides are parallel to the coordinate hyperplanes. We show that these probabilities decay exponentially fast with $s$, when $s$ grows to infinity. Moreover, we prove an associated large deviation principle of speed $s$ for $\\tau /s$ and $\\phi /s$, and we improve the conditions required to obtain the law of large numbers for these variables.",
  "revisions": [
    {
      "version": "v2",
      "updated": "2009-07-03T13:36:56.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "60K35"
    ],
    "keywords": [
      "large numbers",
      "lower large deviations",
      "maximal flow",
      "standard first passage percolation model",
      "coordinate hyperplanes"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 39,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2008arXiv0801.0967R"
    }
  }
}