Infinite Orders and Non-$D$-finite Property of $3$-Dimensional Lattice Walks

TL;DR

This paper confirms that many 3D lattice walk models with large associated groups have infinite groups and explores the non-$D$-finite nature of their generating functions, advancing understanding of their algebraic complexity.

Contribution

It proves the conjecture that models with large groups have infinite groups and investigates the non-$D$-finiteness of their generating functions.

Findings

Many models with large groups have infinite groups.

The generating functions for some models are non-$D$-finite.

Confirmed conjectures about group size and model properties.

Abstract

Recently, Bostan and his coauthors investigated lattice walks restricted to the non-negative octant $\mathbb{N}^3$. For the $35548$ non-trivial models with at most six steps, they found that many models associated to a group of order at least $200$ and conjectured these groups were in fact infinite groups. In this paper, we first confirm these conjectures and then consider the non-$D$-finite property of the generating function for some of these models.