Walks with small steps in the quarter plane

TL;DR

This paper classifies and solves various lattice walk models confined to the first quadrant, revealing which have algebraic or D-finite generating functions and conjecturing the nature of more complex cases.

Contribution

It provides a unified approach to analyze 79 models of quadrant walks, identifying finite groups and solving most with algebraic or D-finite generating functions.

Findings

23 models have finite associated groups

22 models with finite groups have D-finite generating functions

Walks with N, E, W, S, SW, NE steps have algebraic generating functions

Abstract

Let S be a subset of {-1,0,1}^2 not containing (0,0). We address the enumeration of plane lattice walks with steps in S, that start from (0,0) and always remain in the first quadrant. A priori, there are 2^8 problems of this type, but some are trivial. Some others are equivalent to a model of walks confined to a half-plane: such models can be solved systematically using the kernel method, which leads to algebraic generating functions. We focus on the remaining cases, and show that there are 79 inherently different problems to study. To each of them, we associate a group G of birational transformations. We show that this group is finite in exactly 23 cases. We present a unified way of solving 22 of the 23 models associated with a finite group. For each of them, the generating function is found to be D-finite. The 23rd model, known as Gessel's walks, has recently been proved by Bostan et al. to have an algebraic (and hence D-finite) solution. We conjecture that the remaining 56 models, associated with an infinite group, have a non-D-finite generating function. Our approach allows us to recover and refine some known results, and also to obtain new results. For instance, we prove that walks with N, E, W, S, SW and NE steps have an algebraic generating function.