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Facets on the convex hull of d-dimensional Brownian and Lévy motion

Julien Randon-Furling*

Florian Wespi†

  • SAMM (EA 4543), Université Paris-1 Panthéon-Sorbonne, Centre Pierre Mendès-France, 90 rue de Tolbiac, 75013 Paris, France

  • IMSV, Universität Bern, Sidlerstrasse 5, 3012 Bern, Switzerland

  • *julien.randon-furling@univ-paris1.fr
  • †florian.wespi@stat.unibe.ch
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Phys. Rev. E 95, 032129 – Published 17 March, 2017

DOI: https://doi.org/10.1103/PhysRevE.95.032129

Abstract

For stationary, homogeneous Markov processes (viz., Lévy processes, including Brownian motion) in dimension d≥3, we establish an exact formula for the average number of (d−1)-dimensional facets that can be defined by d points on the process's path. This formula defines a universality class in that it is independent of the increments' distribution, and it admits a closed form when d=3, a case which is of particular interest for applications in biophysics, chemistry, and polymer science. We also show that the asymptotical average number of facets behaves as 〈FT(d)〉∼2lnT/Δtd−1, where T is the total duration of the motion and Δt is the minimum time lapse separating points that define a facet.

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Phys. Rev. E 95, 032129– Published 17 March, 2017
  • Received 17 January 2017

©2017 American Physical Society

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