# Surprising variants of Cauchy's formula for mean chord length

## TL;DR

This paper explores random walks on lattices and their mean lengths, revealing variants of Cauchy's classical formula and providing insights into the surprising robustness of the mean chord length relation.

## Contribution

It introduces new variants of Cauchy's mean chord length formula through analysis of lattice-based random walks, offering a simple lattice-based perspective.

## Key findings

- Mean walk length equals lattice size N
- Length distribution scales as n^{-1.5} for large n
- Cauchy's formula holds surprisingly well for irregular paths

## Abstract

We examine isotropic and anisotropic random walks which begin on the surface of linear ($N$), square ($N \times N$), or cubic ($N \times N \times N$) lattices and end upon encountering the surface again. The mean length of walks is equal to $N$ and the distribution of lengths $n$ generally scales as $n^{-1.5}$ for large $n$. Our results are interesting in the context of an old formula due to Cauchy that the mean length of a chord though a convex body of volume $V$ and surface $S$ is proportional to $V/S$. It has been realized in recent years that Cauchy's formula holds surprisingly even if chords are replaced by irregular insect paths or trajectories of colliding gas molecules. The random walk on a lattice offers a simple and transparent understanding of this result in comparison to other formulations based on Boltzmann's transport equation in continuum.