We study the polygons governing the convex hull of a point set created by the steps of $n$ independent two-dimensional random walkers. Each such walk consists of $T$ discrete time steps, where $x$ and $y$ increments are independent and identically distributed Gaussian. We analyze area $A$ and perimeter $L$ of the convex hulls. We obtain probability densities for these two quantities over a large range of the support by using a large-deviation approach allowing us to study densities below ${10}^{-900}$. We find that the densities exhibit in the limit $T\to \infty$ a time-independent scaling behavior as a function of $A/T$ and $L/\sqrt{T}$, respectively. As in the case of one walker $\left(n=1\right)$, the densities follow Gaussian distributions for $L$ and $\sqrt{A}$, respectively. We also obtained the rate functions for the area and perimeter, rescaled with the scaling behavior of their maximum possible values, and found limiting functions for $T\to \infty$, revealing that the densities follow the large-deviation principle. These rate functions can be described by a power law for $n\to \infty$ as found in the $n=1$ case. We also investigated the behavior of the averages as a function of the number of walks $n$ and found good agreement with the predicted behavior.

• Received 24 May 2016

DOI:https://doi.org/10.1103/PhysRevE.94.052120