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

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.