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 analyse (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 , respecively. 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.

Large deviations of convex hulls of self-avoiding random walks

“…Concerning the full distributions, no exact analytical results are available. Here sophisticated large-deviation simulations were used to numerically explore a large part of the full distribution, i.e., down to probabilities far smaller than 10 −100 [21][22][23].…”

“…To fill this void, we use Markov chain Monte Carlo sampling to obtain the distributions of some quantities of inter-arXiv:1804.02371v1 [cond-mat.stat-mech] 6 Apr 2018 est over their whole support. To connect to previous studies [21][22][23] we also compare the aforementioned variants to the standard random walk on a square lattice (LRW). We are mainly interested in the full distribution of the area A and the perimeter L of d = 2 dimensional hulls for walks in the plane, since the effects of the selfinteractions are stronger in lower dimensions.…”

Revisiting random polygonal lines iteratively generated on the plane

“…The properties of the convex hull of a planar Brownian motion is being investigated since the work of Lévy . Recently, interest in studying the characteristics of the convex hull of a planar Brownian motion has significantly increased, namely induced by applications issues . Hence, the expected value of the area of the convex hull of a standard planar Brownian motion is and the expected value of the perimeter is .…”

“…The properties of the convex hull of a planar Brownian motion is being investigated since the work of Lévy. 8 Recently, interest in studying the characteristics of the convex hull of a planar Brownian motion has significantly increased, [9][10][11][12] namely induced by applications issues. 13,14 Hence, 10,15 the expected value of the area of the convex hull of a standard planar…”

Large Deviations of Convex Hulls of the “True” Self-Avoiding Random Walk

“…When asking for more, i.e., for the full distributions, no exact analytical results are available. This motivated the numerical study of the full distributions-or at least large parts of the support-using largedeviation sampling techniques to sample even far into the tails of standard random walks [23] and multiple random walks [24], also in higher dimensions [25]. These numerical studies are rather expensive, since they usually require Markov chain Monte Carlo simulations, allowing one to measure the distribution in regions where the probabilities are as small as 10 −100 .…”