Convex Hull of N Planar Brownian Motions: Exact Results and an Application to Ecology
Abstract
We compute exactly the mean perimeter and area of the convex hull of N independent planar Brownian paths each of duration T, both for open and closed paths. We show that the mean perimeter ⟨L_{N}⟩=α_{N}T and the mean area ⟨A_{N}⟩=β_{N}T for all T. The prefactors α_{N} and β_{N}, computed exactly for all N, increase very slowly (logarithmically) with increasing N. This slow growth is a consequence of extreme value statistics and has interesting implications in an ecological context in estimating the home range of a herd of animals with a population size N.
 Publication:

Physical Review Letters
 Pub Date:
 October 2009
 DOI:
 10.1103/PhysRevLett.103.140602
 arXiv:
 arXiv:0907.0921
 Bibcode:
 2009PhRvL.103n0602R
 Keywords:

 05.40.Fb;
 02.50.r;
 87.23.Cc;
 Random walks and Levy flights;
 Probability theory stochastic processes and statistics;
 Population dynamics and ecological pattern formation;
 Condensed Matter  Statistical Mechanics;
 Mathematical Physics;
 Mathematics  Probability
 EPrint:
 4 pages Revtex, 4 figures included, published version