On the critical exponents of generalized ballot sequences in three dimensions and large tandem walks

TL;DR

This paper investigates the asymptotic behavior of three-dimensional generalized ballot sequences and large tandem walks, demonstrating their non-D-finiteness and utilizing advanced lattice path techniques to solve longstanding problems.

Contribution

It introduces a generalization of tandem walks to larger steps and establishes their bijection with three-dimensional ballot walks, addressing open questions on asymptotics.

Findings

Proves these models are not D-finite.

Provides asymptotic analysis of generalized ballot sequences.

Establishes a bijection between large tandem walks and ballot walks.

Abstract

We answer some questions on the asymptotics of ballot walks raised in [Personal Journal Shalosh B Ekhad and Doron Zeilberger, Apr 5, 2021; see also arXiv:2104.01731] and prove that these models are not D-finite. This short note demonstrates how the powerful tools developed in the last decades on lattice paths in convex cones help us to answer some challenging problems that were out of reach for a long time. On the way we generalize tandem walks to the family of large tandem walks whose steps are of arbitrary length and map them bijectively to a generalization of ballot walks in three dimensions.