Densities of short uniform random walks in higher dimensions

TL;DR

This paper investigates the properties of short uniform random walks in higher dimensions, providing explicit formulas for moments and densities up to five steps, and extending known 2D results to arbitrary dimensions.

Contribution

It offers explicit hypergeometric evaluations of moments and densities for short walks and generalizes key 2D results to higher dimensions.

Findings

Explicit formulas for up to five-step walks.

Complete extensions of 2D results to arbitrary dimensions.

Surprising generalizations of known properties.

Abstract

We study arithmetic properties of short uniform random walks in arbitrary dimensions, with a focus on explicit (hypergeometric) evaluations of the moment functions and probability densities in the case of up to five steps. Somewhat to our surprise, we are able to provide complete extensions to arbitrary dimensions for most of the central results known in the two-dimensional case.