Counting Walks in the Quarter Plane

TL;DR

This paper investigates the nature of generating functions for lattice walks confined to the first quadrant, providing new proofs for algebraicity and holonomy criteria directly from their functional equations.

Contribution

It offers new proofs for classical results on the algebraic and holonomic nature of generating functions of quadrant walks, using a direct approach from their functional equations.

Findings

Reproved Kreweras' algebraicity result for specific walk models.

Provided a new proof of the holonomy criterion for generating functions.

Demonstrated the use of functional equations to analyze series properties.

Abstract

We study planar walks that start from a given point (i\_0, j\_0), take their steps in a finite set S, and are confined in the first quadrant of the plane. Their enumeration can be attacked in a systematic way: the generating function Q(x, y, t) that counts them by their length (variable t) and the coordinates of their endpoint (variables x, y) satisfies a linear functional equation encoding the step-by-step description of walks. For instance, for the square lattice walks starting from the origin, this equation reads (xy-t(x + y + x^2 y + x y^2)) Q(x, y, t) = xy-xtQ(x, 0, t)-ytQ(O, y, t). The central question addressed in this paper is the nature of the series Q(x, y , t). When is it algebraic? When is it D-finite (or holonomic)? Can these properties be derived from the functional equation itself? Our first result is a new proof of an old theorem due to Kreweras, according to which one of these walk models has, for mysterious reasons, an algebraic generating function. Then, we provide a new proof of a holonomy criterion recently proved by M. Petkovsek and the author. In both cases, we work directly from the functional equation.