Short walk adventures

TL;DR

This paper reviews recent advances in short uniform random walks, exploring their connections to Mahler measures and modular forms, and extends techniques to evaluate complex Mahler measures in hypergeometric terms.

Contribution

It introduces new probabilistic methods for analyzing variations of random walks and reduces complex Mahler measures to hypergeometric functions, supporting conjectures about their relation to modular forms.

Findings

Connected random walks to Mahler measures and modular parametrisation.

Extended probabilistic techniques to new random walk variations.

Reduced complex Mahler measures to hypergeometric functions.

Abstract

We review recent development of short uniform random walks, with a focus on its connection to (zeta) Mahler measures and modular parametrisation of the density functions. Furthermore, we extend available "probabilistic" techniques to cover a variation of random walks and reduce some three-variable Mahler measures, which are conjectured to evaluate in terms of $L$-values of modular forms, to hypergeometric form.