{
  "id": "0802.1019",
  "version": "v2",
  "published": "2008-02-07T16:47:12.000Z",
  "updated": "2008-07-08T11:58:21.000Z",
  "title": "On the distribution of the free path length of the linear flow in a honeycomb",
  "authors": [
    "Florin P. Boca",
    "Radu N. Gologan"
  ],
  "comment": "20 pages, 9 figures",
  "categories": [
    "math.DS",
    "math.NT"
  ],
  "abstract": "Let $\\ell \\geq 2$ be an integer. For each $\\eps >0$ remove from $\\R^2$ the union of discs of radius $\\eps$ centered at the integer lattice points $(m,n$, with $m\\nequiv n\\mod{\\ell}$. Consider a point-like particle moving linearly at unit speed, with velocity $\\omega$, along a trajectory starting at the origin, and its free path length $\\tau_{\\ell,\\eps} (\\omega)\\in [0,\\infty]$. We prove the weak convergence of the probability measures associated with the random variables $\\eps \\tau_{\\ell,\\eps}$ as $\\eps \\to 0^+$ and explicitly compute the limiting distribution. For $\\ell=3$ this leads to an asymptotic formula for the length of the trajectory of a billiard in a regular hexagon, starting at the center, with circular pockets of radius $\\eps\\to 0^+$ removed from the corners. For $\\ell=2$ this corresponds to the trajectory of a billiard in a unit square with circular pockets removed from the corners and trajectory starting at the center of the square. The limiting probability measures on $[0,\\infty)$ have a tail at infinity, which contrasts with the case of a square with pockets and trajectory starting from one of the corners, where the limiting probability measure has compact support.",
  "revisions": [
    {
      "version": "v2",
      "updated": "2008-07-08T11:58:21.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "37A60",
      "11B57",
      "11L05",
      "11P21",
      "37D50",
      "82C40",
      "82D05"
    ],
    "keywords": [
      "free path length",
      "linear flow",
      "limiting probability measure",
      "distribution",
      "circular pockets"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 20,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2008arXiv0802.1019B"
    }
  }
}