title: Mean perimeter of the convex hull of a random walk in a semi-infinite medium.
abstract: We study various properties of the convex hull of a planar Brownian motion, defined as the minimum convex polygon enclosing the trajectory, in the presence of an infinite reflecting wall. Recently [Phys. Rev. E 91, 050104(R) (2015)], we announced that the mean perimeter of the convex hull at time t, rescaled by √Dt, is a nonmonotonous function of the initial distance to the wall. In this article, we first give all the details of the derivation of this mean rescaled perimeter, in particular its value when starting from the wall and near the wall. We then determine the physical mechanism underlying this surprising nonmonotonicity of the mean rescaled perimeter by analyzing the impact of the wall on two complementary parts of the convex hull. Finally, we provide a further quantification of the convex hull by determining the mean length of the portion of the reflecting wall visited by the Brownian motion as a function of the initial distance to the wall.
authors.affiliation: Laboratoire de Physique Théorique de la Matière Condensée (UMR CNRS 7600), Université Pierre et Marie Curie, 4 Place Jussieu, 75255 Paris Cedex, France.
authors.forename: Marie
authors.initials: M
authors.lastname: Chupeau
authors.affiliation: Laboratoire de Physique Théorique de la Matière Condensée (UMR CNRS 7600), Université Pierre et Marie Curie, 4 Place Jussieu, 75255 Paris Cedex, France.
authors.forename: Olivier
authors.initials: O
authors.lastname: Bénichou
authors.affiliation: CNRS, Laboratoire de Physique Théorique et Modèles Statistiques, Université Paris-Sud, 91405 Orsay Cedex, France.
authors.forename: Satya N
authors.initials: SN
authors.lastname: Majumdar
mesh_terms: D003198:Computer Simulation; D008962:Models, Theoretical; D009038:Motion