Convex hull of a Brownian motion in confinement

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 one side. We show that the mean perimeter displays a surprising minimum with respect to the starting distance to the wall and exhibits a nonanalyticity 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 to the one-dimensional case, where the mean span is an increasing analytical function. The nonmonotonicity in the two-dimensional 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.

Figure 1

What is the influence of confinement on the convex hull of a 2D Brownian motion? Considering the minimal model of a reflecting confining infinite wall [see (a)], we show that the mean perimeter of the convex hull turns out to be a nonmonotonic function of the initial distance $d$ to the wall [see (b)]. This is in striking contrast to the monotonic behavior of the mean span, which is the 1D analog of the mean perimeter of the convex hull [see (c)]. (a) Convex hull of a Brownian trajectory of starting point O and diffusion coefficient $D$. (b) Mean rescaled perimeter $\langle L_{2\mathrm{D}}^{\left(d\right)}\left(t\right)/\sqrt{Dt}\rangle$ of the convex hull as a function of the rescaled distance to the wall $x=d/\sqrt{Dt}$: analytical (solid line) and simulation (dots) results (see Appendix pp3 for details). The inset presents the different geometrical parameters involved in the calculation. (c) Mean rescaled span $\langle L_{1\mathrm{D}}^{\left(d\right)}\left(t\right)/\sqrt{Dt}\rangle$ of a (semi)confined 1D Brownian motion as a function of the rescaled distance $x=d/\sqrt{Dt}$ to the reflecting point.

Figure 2

Mean span in direction $\theta$ as a function of the rescaled distance $x$ for different values of $\theta$. Like the mean perimeter of the convex hull, this quantity displays a minimum with respect to $x$ for all directions except parallel or perpendicular to the wall.