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, in a Rapid Communication [Phys. Rev. E \textbf{91}, 050104(R) (2015)], we announced that the mean perimeter of the convex hull at time $t$, rescaled by $\sqrt{Dt}$, is a non-monotonous function of the initial distance to the wall. In the present 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 non-monotonicity 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.

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Cited by in corpus (7)

• The convex hull of the run-and-tumble particle in a plane
• Mean perimeter and area of the convex hull of a planar Brownian motion in the presence of resetting
• Mean area of the convex hull of a run and tumble particle in two dimensions
• Extremal statistics for a one-dimensional Brownian motion with a reflective boundary
• Generalized arcsine laws for a sluggish random walker with subdiffusive growth
• First-passage and extreme value statistics for overdamped Brownian motion in a linear potential
• Facets on the convex hull of $d$-dimensional Brownian and Lévy motion