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

Abstract

We examine isotropic and anisotropic random walks which begin on the surface of linear (N), square (N×N), or cubic (N×N×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 through 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.