Galoisian structure of large steps walks in the quadrant

TL;DR

This paper introduces a Galoisian framework for analyzing large steps walks in the quarter plane, extending the group concept and proving algebraicity for new models, including weighted large steps.

Contribution

It extends the group of the walk to large steps models using Galois theory, enabling algebraicity proofs and discovery of new algebraic models.

Findings

Proved algebraicity of weighted large steps models.

Generalized invariants and decoupling in the Galoisian setting.

Validated a conjecture by Bostan, Bousquet-Mélou, and Melczer.

Abstract

The enumeration of walks in the quarter plane confined in the first quadrant has attracted a lot of attention over the past fifteenth years. The generating functions associated to small steps models satisfy a functional equation in two catalytic variables. For such models, Bousquet-M\'elou and Mishna defined a group called the group of the walk which turned out to be central in the classification of small steps models. In particular, its action on the catalytic variables yields a set of change of variables compatible with the structure of the functional equation. This particular set called the orbit has been generalized to models with arbitrary large steps by Bostan, Bousquet-M\'elou and Melczer. 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 group of the walk to models with large steps. Within this Galoisian framework, we generalized the notions of invariants and decoupling. This enable us to develop a general strategy to prove the algebraicity of models with small backward steps. Our constructions lead to the first proofs of algebraicity of weighted models with large steps, proving in particular a conjecture of Bostan, Bousquet-M\'elou and Melczer, and allowing us to find new algebraic models with large steps.