title: Convex hulls of random walks: Large-deviation properties.
abstract: 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.
authors.affiliation: Institut für Physik, Universität Oldenburg, Carl-von-Ossietzky-Straße 9-11, 26111 Oldenburg, Germany.
authors.forename: Gunnar
authors.initials: G
authors.lastname: Claussen
authors.affiliation: Institut für Physik, Universität Oldenburg, Carl-von-Ossietzky-Straße 9-11, 26111 Oldenburg, Germany.
authors.forename: Alexander K
authors.initials: AK
authors.lastname: Hartmann
authors.affiliation: Laboratoire de Physique Théorique et Modèles Statistiques (UMR 8626 du CNRS), Université de Paris-Sud, Bâtiment 100, 91405 Orsay Cedex, France.
authors.forename: Satya N
authors.initials: SN
authors.lastname: Majumdar