Convex hull of N planar Brownian motions: exact results and an application to ecology.
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)>=alpha N sqrt[T] and the mean area <A(N)>=beta(N)T for all T. The prefactors alpha N and beta 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.
Randon-Furling
Julien
J
Laboratoire de Physique Théorique et Modèles Statistiques (UMR CNRS 8626) Université Paris-Sud, Bâtiment 100, 91405 Orsay Cedex, France.
Majumdar
Satya N
SN
Comtet
Alain
A