On the finiteness of the group associated with weighted walks in multidimensional orthants

TL;DR

This paper investigates the conditions under which the group associated with weighted walks in multidimensional orthants is finite, providing classifications and characterizations in various dimensions to aid in computing generating functions.

Contribution

It offers a complete characterization of parameters for finiteness in 2D, shows higher-dimensional groups are reflection groups, and classifies 3D cases with Weyl properties.

Findings

Complete characterization of finiteness in 2D

Higher-dimensional groups are reflection groups when finite

Classification of 3D parameters with Weyl property

Abstract

In the study of walks with small steps confined to multidimensional orthants, a certain group of transformations plays a central role. In particular, several techniques to potentially compute the generating function, including the orbit sum method, can only be applied when this group is finite. In this note, we present three new results concerning this group. First, in two dimensions, we provide a complete characterization of the weight parameters that yield a finite group. In higher dimensions, we show that whenever the group is finite, it must necessarily be isomorphic to a simpler reflection group. Finally, in dimension three, we give a full classification of the parameters leading to a finite group that also satisfies an additional Weyl property.