# Mean perimeter and mean area of the convex hull over planar random walks

## TL;DR

This paper analyzes the geometric properties of convex hulls formed by planar random walks, deriving asymptotic behaviors for mean perimeter and area, with applications in biology and ecology.

## Contribution

It provides the first derivation of the large $n$ asymptotics of the mean perimeter and area of convex hulls for planar random walks with symmetric jumps, including finite-size corrections.

## Key findings

- Mean perimeter asymptotics derived for large n
- Mean area computed for isotropic Gaussian jumps
- Finite-size corrections improve accuracy for small n

## Abstract

We investigate the geometric properties of the convex hull over $n$ successive positions of a planar random walk, with a symmetric continuous jump distribution. We derive the large $n$ asymptotic behavior of the mean perimeter. In addition, we compute the mean area for the particular case of isotropic Gaussian jumps. While the leading terms of these asymptotics are universal, the subleading (correction) terms depend on finer details of the jump distribution and describe a "finite size effect" of discrete-time jump processes, allowing one to accurately compute the mean perimeter and the mean area even for small $n$, as verified by Monte Carlo simulations. This is particularly valuable for applications dealing with discrete-time jumps processes and ranging from the statistical analysis of single-particle tracking experiments in microbiology to home range estimations in ecology.