{
  "id": "0907.0614",
  "version": "v1",
  "published": "2009-07-03T12:27:21.000Z",
  "updated": "2009-07-03T12:27:21.000Z",
  "title": "Upper large deviations for maximal flows through a tilted cylinder",
  "authors": [
    "Marie Theret"
  ],
  "comment": "14 pages, 4 figures",
  "categories": [
    "math.PR"
  ],
  "abstract": "We consider the standard first passage percolation model in $\\ZZ^d$ for $d\\geq 2$ and we study the maximal flow from the upper half part to the lower half part (respectively from the top to the bottom) of a cylinder whose basis is a hyperrectangle of sidelength proportional to $n$ and whose height is $h(n)$ for a certain height function $h$. We denote this maximal flow by $\\tau_n$ (respectively $\\phi_n$). We emphasize the fact that the cylinder may be tilted. We look at the probability that these flows, rescaled by the surface of the basis of the cylinder, are greater than $\\nu(\\vec{v})+\\eps$ for some positive $\\eps$, where $\\nu(\\vec{v})$ is the almost sure limit of the rescaled variable $\\tau_n$ when $n$ goes to infinity. On one hand, we prove that the speed of decay of this probability in the case of the variable $\\tau_n$ depends on the tail of the distribution of the capacities of the edges: it can decays exponentially fast with $n^{d-1}$, or with $n^{d-1} \\min(n,h(n))$, or at an intermediate regime. On the other hand, we prove that this probability in the case of the variable $\\phi_n$ decays exponentially fast with the volume of the cylinder as soon as the law of the capacity of the edges admits one exponential moment; the importance of this result is however limited by the fact that $\\nu(\\vec{v})$ is not in general the almost sure limit of the rescaled maximal flow $\\phi_n$, but it is the case at least when the height $h(n)$ of the cylinder is negligible compared to $n$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2009-07-03T12:27:21.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "60K35"
    ],
    "keywords": [
      "maximal flow",
      "upper large deviations",
      "tilted cylinder",
      "standard first passage percolation model",
      "decays exponentially fast"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 14,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2009arXiv0907.0614T"
    }
  }
}