Convex Hulls of Random Walks: Large-Deviation Properties

TL;DR

This paper investigates the full probability distributions of the perimeter and area of convex hulls formed by two-dimensional random walks, revealing universal scaling laws and rare event behaviors through advanced large-deviation numerical methods.

Contribution

It introduces a numerical large-deviation approach to analyze the full distributions of convex hull observables for random walks, including rare events and universal scaling laws.

Findings

Distributions exhibit universal scaling with respect to T.

Rate functions show linear and quadratic dependence for perimeter and area.

Rare event probabilities as small as 10^{-300} are computed.

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/\sqrt{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.