Convex hull of a Brownian motion in confinement

M. Chupeau 1 O. Bénichou 1 S. N. Majumdar 2

Physical Review E : Statistical, Nonlinear, and Soft Matter Physics, American Physical Society, 2015, 91, pp.050104

We study the effect of confinement on the mean perimeter of the convex hull of a planar Brownian motion, defined as the minimum convex polygon enclosing the trajectory. We use a minimal model where an infinite reflecting wall confines the walk to its one side. We show that the mean perimeter displays a surprising minimum with respect to the starting distance to the wall and exhibits a non-analyticity for small distances. In addition, the mean span of the trajectory in a fixed direction {$\theta \in ]0,\pi/2[$}, which can be shown to yield the mean perimeter by integration over $\theta$, presents these same two characteristics. This is in striking contrast with the one dimensional case, where the mean span is an increasing analytical function. The non-monotonicity in the 2D case originates from the competition between two antagonistic effects due to the presence of the wall: reduction of the space accessible to the Brownian motion and effective repulsion.

• 1. LPTMC - Laboratoire de Physique Théorique de la Matière Condensée
• 2. LPTMS - Laboratoire de Physique Théorique et Modèles Statistiques