Journal of Statistical Physics

, Volume 138, Issue 6, pp 955–1009

Random Convex Hulls and Extreme Value Statistics

  • Satya N. Majumdar
  • Alain Comtet
  • Julien Randon-Furling
Article

DOI: 10.1007/s10955-009-9905-z

Cite this article as:
Majumdar, S.N., Comtet, A. & Randon-Furling, J. J Stat Phys (2010) 138: 955. doi:10.1007/s10955-009-9905-z

Abstract

In this paper we study the statistical properties of convex hulls of N random points in a plane chosen according to a given distribution. The points may be chosen independently or they may be correlated. After a non-exhaustive survey of the somewhat sporadic literature and diverse methods used in the random convex hull problem, we present a unifying approach, based on the notion of support function of a closed curve and the associated Cauchy’s formulae, that allows us to compute exactly the mean perimeter and the mean area enclosed by the convex polygon both in case of independent as well as correlated points. Our method demonstrates a beautiful link between the random convex hull problem and the subject of extreme value statistics. As an example of correlated points, we study here in detail the case when the points represent the vertices of n independent random walks. In the continuum time limit this reduces to n independent planar Brownian trajectories for which we compute exactly, for all n, the mean perimeter and the mean area of their global convex hull. Our results have relevant applications in ecology in estimating the home range of a herd of animals. Some of these results were announced recently in a short communication [Phys. Rev. Lett. 103:140602, 2009].

Convex hull Brownian motion Random walks 

Copyright information

© Springer Science+Business Media, LLC 2009

Authors and Affiliations

  • Satya N. Majumdar
    • 1
  • Alain Comtet
    • 1
    • 2
  • Julien Randon-Furling
    • 1
  1. 1.LPTMSUniv. Paris-Sud 11Orsay CedexFrance
  2. 2.Université Pierre et Marie Curie-Paris 6Paris Cedex 05France
  3. 3.AG Rieger, Theoretische PhysikUniversität des Saarlandes, PF 151150SaarbrückenGermany