Convex hull of a Brownian motion in confinement

Abstract

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.

References in corpus (5)

• Persistence and First-Passage Properties in Non-equilibrium Systems
• Spatial extent of an outbreak in animal epidemics
• Volumetric Properties of the Convex Hull of an n-dimensional Brownian Motion
• Convex hull of n planar Brownian paths: an exact formula for the average number of edges
• Universality and time-scale invariance for the shape of planar Lévy processes

Cited by in corpus (12)

• Extreme value statistics of correlated random variables: a pedagogical review
• Mean perimeter and area of the convex hull of a planar Brownian motion in the presence of resetting
• Unravelling intermittent features in single particle trajectories by a local convex hull method
• Mean perimeter and mean area of the convex hull over planar random walks
• Convex Hulls of Random Walks in Higher Dimensions: A Large Deviation Study
• Convex Hulls of Multiple Random Walks: A Large-Deviation Study
• Exact distributions of cover times for $N$ independent random walkers in one dimension
• Mean perimeter of the convex hull of a random walk in a semi-infinite medium
• Extreme-value statistics from Lagrangian convex hull analysis for homogeneous turbulent Boussinesq convection and MHD convection
• Generalized arcsine laws for a sluggish random walker with subdiffusive growth
• Large deviations of a random walk model with emerging territories
• Facets on the convex hull of $d$-dimensional Brownian and Lévy motion