{
  "id": "1605.06958",
  "version": "v1",
  "published": "2016-05-23T09:49:09.000Z",
  "updated": "2016-05-23T09:49:09.000Z",
  "title": "Convex Hulls of Multiple Random Walks: A Large-Deviation Study",
  "authors": [
    "Timo Dewenter",
    "Gunnar Claussen",
    "Alexander K. Hartmann",
    "Satya N. Majumdar"
  ],
  "comment": "11 pages, 14 figures",
  "categories": [
    "cond-mat.stat-mech",
    "physics.data-an"
  ],
  "abstract": "We study the polygons governing the convex hull of a point set created by the steps of $n$ independent two-dimensional random walkers. Each such walk consists of $T$ discrete time steps, where $x$ and $y$ increments are i.i.d. Gaussian. We analyze area $A$ and perimeter $L$ of the convex hulls. We obtain probability densities for these two quantities over a large range of the support by using a large-deviation approach allowing us to study densities below $10^{-900}$. We find that the densities exhibit a universal scaling behavior as a function of $A/T$ and $L/\\sqrt{T}$, respectively. As in the case of one walker ($n=1$), the densities follow Gaussian distributions for $L$ and $\\sqrt{A}$, respectively. We also obtained the rate functions for the area and perimeter, rescaled with the scaling behavior of their maximum possible values, and found limiting functions for $T \\rightarrow \\infty$, revealing that the densities follow the large-deviation principle. These rate functions can be described by a power law for $n \\rightarrow \\infty$ as found in the $n=1$ case. We also investigated the behavior of the averages as a function of the number of walks $n$ and found good agreement with the predicted behavior.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2016-05-23T09:49:09.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "multiple random walks",
      "convex hull",
      "large-deviation study",
      "independent two-dimensional random walkers",
      "rate functions"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 11,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}