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

Andrew Elvey Price, Emmanuel Humbert, Kilian Raschel

TL;DR

The paper studies finiteness of the combinatorial group $G$ associated with weighted walks in the orthant, a property that enables orbit-sum methods to derive generating functions. It proves a complete 2D weight-classification for finiteness, shows any finite $G$ in higher dimensions is isomorphic to a reflection group $H$, and in 3D provides a full Weyl-property classification via Coxeter data, with reductions to 2D analysis. The results connect finiteness to reflection/Coxeter theory and supply explicit model inventories and central weightings that realize finite groups. Together, they map exactly which cone-walk models admit algebraic or D-finite generating functions under orbit-sum techniques and guide extensions to higher dimensions.

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.

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

TL;DR

The paper studies finiteness of the combinatorial group associated with weighted walks in the orthant, a property that enables orbit-sum methods to derive generating functions. It proves a complete 2D weight-classification for finiteness, shows any finite in higher dimensions is isomorphic to a reflection group , and in 3D provides a full Weyl-property classification via Coxeter data, with reductions to 2D analysis. The results connect finiteness to reflection/Coxeter theory and supply explicit model inventories and central weightings that realize finite groups. Together, they map exactly which cone-walk models admit algebraic or D-finite generating functions under orbit-sum techniques and guide extensions to higher dimensions.

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.
Paper Structure (9 sections, 6 theorems, 30 equations, 1 figure)

This paper contains 9 sections, 6 theorems, 30 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Main results
  3. The combinatorial group
  4. A new complete classification in dimension two
  5. A reflection group
  6. Classification of finite groups in three dimensions and beyond
  7. Proof outline for Theorem \ref{['thm:1']}
  8. Proof of Theorem \ref{['thm:2']}
  9. Proof outline for Theorem \ref{['thm:3']}

Key Result

Theorem 1

In dimension $2$, the only possible orders of $G$ are $4$, $6$, $8$ and $10$.

Figures (1)

  • Figure 1: Examples of finite group models taken from KaYa-15. From left to right, a model with a group of order $4$, $6$, $8$ and $10$. They should be normalised to satisfy the property that $\sum_{(i,j)\in\mathcal{S}}w(i,j)=1$. The paper KaYa-15 actually contains infinite families of finite group examples. For example, the third example defines a group model of order $8$ if and only if the associated weights $w(i,j)$ satisfy $w(1,0)w(-1,0)=w(1,1)w(-1,-1)\neq 0$.

Theorems & Definitions (7)

  • Theorem 1
  • Theorem 2
  • Theorem 3
  • Lemma 1
  • proof
  • Proposition 1
  • Proposition 2