The convex hull of a planar random walk: perimeter, diameter, and shape

TL;DR

This paper investigates the asymptotic properties of the convex hull of a planar random walk, analyzing perimeter, diameter, and shape behavior under different drift conditions with new limit theorems.

Contribution

It provides novel asymptotic results and distributional limit theorems for the convex hull's perimeter and diameter, including shape approximation in the zero-drift case.

Findings

For non-zero drift, perimeter-to-diameter ratio converges to 2 almost surely.

In zero-drift, the convex hull shape can approximate any convex set with diameter 1.

The ratio of perimeter to diameter oscillates between 2 and π almost surely.

Abstract

We study the convex hull of the first $n$ steps of a planar random walk, and present large-$n$ asymptotic results on its perimeter length $L_n$, diameter $D_n$, and shape. In the case where the walk has a non-zero mean drift, we show that $L_n / D_n \to 2$ a.s., and give distributional limit theorems and variance asymptotics for $D_n$, and in the zero-drift case we show that the convex hull is infinitely often arbitrarily well-approximated in shape by any unit-diameter compact convex set containing the origin, and then $\liminf_{n \to \infty} L_n/D_n =2$ and $\limsup_{n \to \infty} L_n /D_n = \pi$, a.s. Among the tools that we use is a zero-one law for convex hulls of random walks.