{
  "id": "1501.01041",
  "version": "v1",
  "published": "2015-01-05T23:54:09.000Z",
  "updated": "2015-01-05T23:54:09.000Z",
  "title": "Convex Hulls of Random Walks: Large-Deviation Properties",
  "authors": [
    "Gunnar Claussen",
    "Alexander K. Hartmann",
    "Satya N. Majumdar"
  ],
  "comment": "10 pages, 12 figures, 2 tables",
  "categories": [
    "cond-mat.stat-mech",
    "physics.data-an"
  ],
  "abstract": "We study the convex hull of the set of points visited by a two-dimensional random walker of T discrete time steps. Two natural observables that characterize the convex hull in two dimensions are its perimeter L and area A. While the mean perimeter <L> and the mean area <A> have been studied before, analytically and numerically, and exact results are known for large T (Brownian motion limit), little is known about the full distributions P(A) and P(L). In this paper, we provide numerical results for these distributions. We use a sophisticated large-deviation approach that allows us to study the distributions over a larger range of the support, where the probabilities P(A) and P(L) are as small as 10^{-300}. We analyze (open) random walks as well as (closed) Brownian bridges on the two-dimensional discrete grid as well as in the two-dimensional plane. The resulting distributions exhibit, for large T, a universal scaling behavior (independent of the details of the jump distributions) as a function of A/T and L/\\sqrt{T}, respectively. We are also able to obtain the rate function, describing rare events at the tails of these distributions, via a numerical extrapolation scheme and find a linear and square dependence as a function of the rescaled perimeter and the rescaled area, respectively.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2015-01-05T23:54:09.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "convex hull",
      "random walks",
      "large-deviation properties",
      "distributions",
      "brownian motion limit"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 10,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2015arXiv150101041C"
    }
  }
}