{
  "id": "0807.4072",
  "version": "v2",
  "published": "2008-07-25T11:35:54.000Z",
  "updated": "2008-09-17T09:04:59.000Z",
  "title": "Sylvester's question and the Random Acceleration Process",
  "authors": [
    "H. J. Hilhorst",
    "P. Calka",
    "G. Schehr"
  ],
  "comment": "29 pages, 4 figures; references added and minor changes",
  "journal": "J. Stat. Mech. (2008) P10010",
  "doi": "10.1088/1742-5468/2008/10/P10010",
  "categories": [
    "cond-mat.stat-mech"
  ],
  "abstract": "Let n points be chosen randomly and independently in the unit disk. \"Sylvester's question\" concerns the probability p_n that they are the vertices of a convex n-sided polygon. Here we establish the link with another problem. We show that for large n this polygon, when suitably parametrized by a function r(phi) of the polar angle phi, satisfies the equation of the random acceleration process (RAP), d^2 r/d phi^2 = f(phi), where f is Gaussian noise. On the basis of this relation we derive the asymptotic expansion log p_n = -2n log n + n log(2 pi^2 e^2) - c_0 n^{1/5} + ..., of which the first two terms agree with a rigorous result due to Barany. The nonanalyticity in n of the third term is a new result. The value 1/5 of the exponent follows from recent work on the RAP due to Gyorgyi et al. [Phys. Rev. E 75, 021123 (2007)]. We show that the n-sided polygon is effectively contained in an annulus of width \\sim n^{-4/5} along the edge of the disk. The distance delta_n of closest approach to the edge is exponentially distributed with average 1/(2n).",
  "revisions": [
    {
      "version": "v2",
      "updated": "2008-09-17T09:04:59.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "random acceleration process",
      "sylvesters question",
      "polar angle phi",
      "asymptotic expansion log",
      "convex n-sided polygon"
    ],
    "tags": [
      "journal article"
    ],
    "publication": {
      "journal": "Journal of Statistical Mechanics: Theory and Experiment",
      "year": 2008,
      "month": "Oct",
      "volume": 2008,
      "number": 10,
      "pages": 10010
    },
    "note": {
      "typesetting": "TeX",
      "pages": 29,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2008JSMTE..10..010H"
    }
  }
}