Mean perimeter of the convex hull of a random walk in a semi-infinite medium

TL;DR

This paper investigates how a reflecting wall influences the mean perimeter of the convex hull of a planar Brownian motion, revealing a non-monotonous dependence on initial distance and analyzing the wall's impact on the hull's structure.

Contribution

It provides a detailed derivation of the mean perimeter of the convex hull in the presence of a reflecting wall and explains the non-monotonous behavior observed.

Findings

Mean perimeter exhibits non-monotonous dependence on initial distance to the wall.

The impact of the wall on different parts of the convex hull is characterized.

The mean length of the wall visited by the Brownian motion is quantified.

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.