Convex Hull of N Planar Brownian Motions: Exact Results and an Application to Ecology

TL;DR

This paper provides exact formulas for the mean perimeter and area of the convex hull of N independent planar Brownian motions, revealing slow logarithmic growth with N and applications in ecological home range estimation.

Contribution

It derives exact expressions for the mean perimeter and area of convex hulls of multiple Brownian paths, highlighting the role of extreme value statistics and ecological implications.

Findings

Mean perimeter scales as ppa_N \, \u221a T with ppa_N increasing logarithmically with N.

Mean area scales as eta_N T, with eta_N also increasing slowly with N.

Results have implications for estimating animal home ranges in 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 > = \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 implication in ecological context in estimating the home range of a herd of animals with population size N.