Convex hulls of random walks: Large-deviation properties

• Published in: Physical review. E, Statistical, nonlinear, and soft matter physics Statistical, nonlinear, and soft matter physics, 2015-05, Vol.91 (5), p.052104-052104, Article 052104
• Main Authors: Claussen, Gunnar, Hartmann, Alexander K, Majumdar, Satya N
• Format: Article
• Language: English
• Subjects: Physics
• ISSN: 1539-3755; 1550-2376
• DOI: 10.1103/PhysRevE.91.052104

We study the convex hull of the set of points visited by a two-dimensional random walker of T discrete time steps. Two natural observables that characterize the convex hull in two dimensions are its perimeter L and area A. While the mean perimeter 〈L〉 and the mean area 〈A〉 have been studied before, analytically and numerically, and exact results are known for large T (Brownian motion limit), little is known about the full distributions P(A) and P(L). In this paper we provide numerical results for these distributions. We use a sophisticated large-deviation approach that allows us to study the distributions over a larger range of the support, where the probabilities P(A) and P(L) are as small as 10(-300). We analyze (open) random walks as well as (closed) Brownian bridges on the two-dimensional discrete grid as well as in the two-dimensional plane. The resulting distributions exhibit, for large T, a universal scaling behavior (independent of the details of the jump distributions) as a function of A/T and L/√[T], respectively. We are also able to obtain the rate function, describing rare events at the tails of these distributions, via a numerical extrapolation scheme and find a linear and square dependence as a function of the rescaled perimeter and the rescaled area, respectively.