Asymptotic probability of a fixed edge being on the boundary of the convex hull of a random walk in $\mathbb{Z}^2$

TL;DR

This paper investigates the asymptotic probability that a fixed edge of a simple symmetric random walk in two-dimensional integer space lies on the convex hull boundary as the number of steps grows large.

Contribution

It provides an analysis of the asymptotic behavior of the probability that a specific edge is on the convex hull boundary in a 2D random walk.

Findings

Asymptotic probability of a fixed edge on the convex hull boundary is characterized.

The probability tends to a specific limit as the number of steps increases.

Results contribute to understanding geometric properties of random walks in lattice spaces.

Abstract

A simple symmetric random walk in the space $\mathbb{Z}^2$ is considered. The asymptotic behavior as the number of jumps tends to infinity of the probability that a fixed edge of the random walk lies in the polygon that forms the boundary of the convex hull is investigated.