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Satya Majumdar (Orsay)

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 < LN> = aN T1/2 and the mean area < AN> = bN T for all T's. The prefactors aN and bN, computed exactly for all N's, increase very slowly (logarithmically) with increasing N. This slow growth is a consequence of extreme value statistics and has interesting implications in ecological context in estimating the home range of a herd of animals with population of size N.