Walks in the Quarter Plane with Multiple Steps

TL;DR

This paper classifies quarter plane walks with multiple steps, identifying families with specific symmetry groups and characterizing their algebraic properties, extending prior models with multiplicities in step directions.

Contribution

It introduces a comprehensive classification of quarter plane walks with multiplicities, revealing infinite families linked to symmetry groups D4, D6, and D8, expanding understanding of their algebraic nature.

Findings

Identified families with groups D4, D6, D8

Covered models with multiplicities 0 to 3

Most models are D-finite, with three exceptions

Abstract

We extend the classification of nearest neighbour walks in the quarter plane to models in which multiplicities are attached to each direction in the step set. Our study leads to a small number of infinite families that completely characterize all the models whose associated group is D4, D6, or D8. These families cover all the models with multiplicites 0, 1, 2, or 3, which were experimentally found to be D-finite --- with three noteworthy exceptions.