Convex hulls of multiple random walks: A large-deviation study.

Abstract

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→∞ a time-independent scaling behavior as a function of A/T and L/sqrt[T], respectively. As in the case of one walker (n=1), 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→∞, revealing that the densities follow the large-deviation principle. These rate functions can be described by a power law for n→∞ 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.