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A Galois structure on the orbit of large steps walks in the quadrant

Pierre Bonnet, Charlotte Hardouin

TL;DR

This work extends the orbit framework for quarter-plane walks from small to large steps by endowing the orbit with a Galois action and developing a systematic theory of Galois invariants and decoupling. The authors show how to translate a two-variable kernel equation into one-variable algebraic problems through invariants and decouplings, yielding an algebraicity proof whenever the orbit is finite. They apply the method to the model $ ext{G}_{\lambda}$, proving algebraicity and giving an explicit degree-$32$ minimal polynomial for $Q(0,0,t)$, thereby resolving a conjecture of BBMM and illustrating a pathway to families of algebraic large-step models. The approach unifies kernel methods, orbit theory, and finite-automorphism groups under a Galois-theoretic lens, with potential broad applicability to large-step walks in cones.

Abstract

The enumeration of weighted walks in the quarter plane reduces to studying a functional equation with two catalytic variables. When the steps of the walk are small, Bousquet-Mélou and Mishna defined a group called the group of the walk which turned out to be crucial in the classification of the small steps models. In particular, its action on the catalytic variables provides a convenient set of changes of variables in the functional equation. This particular set called the orbit has been generalized to models with arbitrary large steps by Bostan, Bousquet-Mélou and Melczer (BBMM). However, the orbit had till now no underlying group. In this article, we endow the orbit with the action of a Galois group, which extends the notion of the group of the walk to models with large steps. As an application, we look into a general strategy to prove the algebraicity of models with small backwards steps, which uses the fundamental objects that are invariants and decoupling. The group action on the orbit allows us to develop a Galoisian approach to these two notions. Up to the knowledge of the finiteness of the orbit, this gives systematic procedures to test their existence and construct them. Our constructions lead to the first proofs of algebraicity of weighted models with large steps, proving in particular a conjecture of BBMM, and allowing to find new algebraic models with large steps.

A Galois structure on the orbit of large steps walks in the quadrant

TL;DR

This work extends the orbit framework for quarter-plane walks from small to large steps by endowing the orbit with a Galois action and developing a systematic theory of Galois invariants and decoupling. The authors show how to translate a two-variable kernel equation into one-variable algebraic problems through invariants and decouplings, yielding an algebraicity proof whenever the orbit is finite. They apply the method to the model , proving algebraicity and giving an explicit degree- minimal polynomial for , thereby resolving a conjecture of BBMM and illustrating a pathway to families of algebraic large-step models. The approach unifies kernel methods, orbit theory, and finite-automorphism groups under a Galois-theoretic lens, with potential broad applicability to large-step walks in cones.

Abstract

The enumeration of weighted walks in the quarter plane reduces to studying a functional equation with two catalytic variables. When the steps of the walk are small, Bousquet-Mélou and Mishna defined a group called the group of the walk which turned out to be crucial in the classification of the small steps models. In particular, its action on the catalytic variables provides a convenient set of changes of variables in the functional equation. This particular set called the orbit has been generalized to models with arbitrary large steps by Bostan, Bousquet-Mélou and Melczer (BBMM). However, the orbit had till now no underlying group. In this article, we endow the orbit with the action of a Galois group, which extends the notion of the group of the walk to models with large steps. As an application, we look into a general strategy to prove the algebraicity of models with small backwards steps, which uses the fundamental objects that are invariants and decoupling. The group action on the orbit allows us to develop a Galoisian approach to these two notions. Up to the knowledge of the finiteness of the orbit, this gives systematic procedures to test their existence and construct them. Our constructions lead to the first proofs of algebraicity of weighted models with large steps, proving in particular a conjecture of BBMM, and allowing to find new algebraic models with large steps.
Paper Structure (11 sections, 8 theorems, 16 equations, 3 figures)

This paper contains 11 sections, 8 theorems, 16 equations, 3 figures.

Table of Contents

  1. Introduction and preliminaries
  2. Walks in the quarter plane
  3. Generating function and classification
  4. The group and the orbit
  5. A strategy based on invariants and decoupling
  6. A Galois structure on the orbit
  7. Construction of invariants and decoupling
  8. Fractions as elements of $k(\mathcal{O})$
  9. Galois invariants
  10. Galois decoupling
  11. An example: the model $\mathcal{G}_{\lambda}$ is algebraic

Key Result

Lemma 9

Let $(F(X,t),G(Y,t))$ be a pair of $t$-invariants. If the coefficients of the power series $\frac{F(X,t)-G(Y,t)}{\widetilde{K}(X,Y,t)} \in \mathbb{C}_{\mathrm{mul}}(X,Y)[[t]]$ have no pole at $X=0$ nor $Y=0$, then there exists a series $A(t)$ in $\mathbb{C}[[t]]$ such that $F(X,t) = G(Y,t) = A(t)$.

Figures (3)

  • Figure 1: An example of weighted model and walk
  • Figure 2: The partial classifications for two families of unweighted models
  • Figure 3: The orbit $\mathcal{O}_{12}$ of $\mathcal{G}_{\lambda}$ in two perspectives, illustrating a distance transitivity property (the $0$-chains $X_i$ and $Y_i$ are the sums of vertices in their respective regions).

Theorems & Definitions (23)

  • Example 1
  • Definition 2: Definition 3.1 in bostan2018counting
  • Example 3
  • Example 4
  • Definition 5
  • Definition 6: Invariants, Def. 2.3 in BousquetMelouThreequadrant
  • Definition 7: Decoupling
  • Example 8
  • Lemma 9: Lemma 2.6 in BousquetMelouThreequadrant
  • Lemma 10: Lemmas 3.7 and 3.8 in BHorb
  • ...and 13 more