{
  "id": "1210.2231",
  "version": "v3",
  "published": "2012-10-08T10:58:58.000Z",
  "updated": "2013-05-06T08:50:11.000Z",
  "title": "Tail asymptotics of free path lengths for the periodic Lorentz process. On Dettmann's geometric conjectures",
  "authors": [
    "Peter Nandori",
    "Domokos Szasz",
    "Tamas Varju"
  ],
  "categories": [
    "math.DS"
  ],
  "abstract": "In the simplest case, consider a $\\mathbb{Z}^d$-periodic ($d \\geq 3$) arrangement of balls of radii < 1/2, and select a random direction and point (outside the balls). According to Dettmann's first conjecture, the probability that the so determined free flight (until the first hitting of a ball) is larger than $t >>1$ is $\\sim \\frac{C}{t}$, where $C$ is explicitly given by the geometry of the model. In its simplest form, Dettmann's second conjecture is related to the previous case with tangent balls (of radii 1/2). The conjectures are established in a more general setup: for $\\mathcal{L}$-periodic configuration of - possibly intersecting - convex bodies with $\\mathcal{L}$ being a non-degenerate lattice. These questions are related to P\\'olya's visibility problem (1918), to theories of Bourgain-Golse-Wennberg (1998-) and of Marklof-Str\\\"{o}mbergsson (2010-). The results also provide the asymptotic covariance of the periodic Lorentz process assuming it has a limit in the super-diffusive scaling, a fact if $d = 2$ and the horizon is infinite.",
  "revisions": [
    {
      "version": "v3",
      "updated": "2013-05-06T08:50:11.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "37D50"
    ],
    "keywords": [
      "periodic lorentz process",
      "free path lengths",
      "dettmanns geometric conjectures",
      "tail asymptotics",
      "dettmanns first conjecture"
    ],
    "publication": {
      "doi": "10.1007/s00220-014-2086-x",
      "journal": "Communications in Mathematical Physics",
      "year": 2014,
      "month": "Oct",
      "volume": 331,
      "number": 1,
      "pages": 111
    },
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2014CMaPh.331..111N"
    }
  }
}