[Paper Review] On the Holonomy or Algebraicity of Generating Functions Counting Lattice Walks in the Quarter-Plane

Guy Fayolle, Kilian Raschel|arXiv (Cornell University)|Apr 10, 2010

Mathematics and ApplicationsMathematics9 references20 citations

TL;DR

This paper establishes the holonomic and algebraic nature of bivariate generating functions for 23 small-step lattice walks in the quarter-plane using direct application of general theorems from [4], particularly focusing on functional equations derived from probabilistic random walk models. It confirms that generating functions are holonomic, with algebraicity occurring in four cases—including Gessel’s walk—based on the structure of the associated group and kernel equations on genus-one Riemann surfaces uniformized by Weierstrass elliptic functions.

ABSTRACT

In two recent works \cite{BMM,BK}, it has been shown that the counting generating functions (CGF) for the 23 walks with small steps confined in a quadrant and associated with a finite group of birational transformations are holonomic, and even algebraic in 4 cases -- in particular for the so-called Gessel's walk. It turns out that the type of functional equations satisfied by these CGF appeared in a probabilistic context almost 40 years ago. Then a method of resolution was proposed in \cite{FIM}, involving at once algebraic tools and a reduction to boundary value problems. Recently this method has been developed in a combinatorics framework in \cite{Ra}, where a thorough study of the explicit expressions for the CGF is proposed. The aim of this paper is to derive the nature of the bivariate CGF by a direct use of some general theorems given in \cite{FIM}.

Motivation & Objective

• To determine the nature—holonomic or algebraic—of bivariate generating functions (CGF) for small-step lattice walks confined to the quarter-plane.
• To resolve the functional equations governing these CGFs using established theorems from [4], particularly those related to boundary value problems and conformal gluing functions.
• To clarify why certain walks yield algebraic CGFs while others are only holonomic, based on group structure and curve genus.
• To provide a direct, systematic derivation of the CGF nature via uniformization of the kernel curve using Weierstrass elliptic functions.

Proposed method

• The paper applies general theorems from [4] on functional equations of the form $ K(x,y)F(x,y,z) = c(x)F(x,0,z) + \tilde{c}(y)F(0,y,z) + c_0(x,y) $, which arise in probabilistic random walk models.
• It uses the group of birational transformations $ W = \langle \xi, \eta \rangle $ associated with the step set $ \mathcal{S} $, which acts on the kernel curve $ K(x,y) = 0 $, to analyze symmetry and monodromy.
• The kernel curve $ K(x,y) = 0 $ is analyzed for genus; when genus is one, it is uniformized via the Weierstrass elliptic function $ \wp $, enabling explicit integral representations.
• The method involves checking whether a certain sum of residues $ \sum_{k=0}^{n-1} \frac{\psi_{\delta^k}}{\prod_{i=1}^k f_{\delta^i}} $ vanishes on the curve $ K(x,y) = 0 $, which determines holonomy or algebraicity.
• For walks with finite group order $ n = 2, 3, 4 $, the paper evaluates this sum explicitly and shows it vanishes only in specific cases, indicating algebraicity.
• The analysis relies on conformal gluing functions and the structure of the jump set $ \mathcal{S} $, particularly symmetry and the form of $ c(x), \tilde{c}(y) $, to classify cases.

Experimental results

Research questions

• RQ1 Under what conditions is the bivariate generating function for a small-step lattice walk in the quarter-plane holonomic or algebraic?
• RQ2 How does the group of birational transformations associated with the step set determine the nature of the generating function?
• RQ3 Why do only four of the 23 walks with finite group yield algebraic generating functions, while the rest are only holonomic?
• RQ4 What role does the genus-one structure of the kernel curve $ K(x,y) = 0 $ play in determining algebraicity via Weierstrass uniformization?
• RQ5 Can the functional equation approach from probabilistic random walks be directly used to derive the algebraic or holonomic nature of the generating function?

Key findings

• The generating function is holonomic for all 23 walks with finite group in the quarter-plane, confirming earlier results from [2] and [11].
• Four walks, including Gessel’s walk, have algebraic generating functions, while the remaining 19 are holonomic but not algebraic.
• Algebraicity occurs precisely when the sum $ \sum_{k=0}^{n-1} \frac{\psi_{\delta^k}}{\prod_{i=1}^k f_{\delta^i}} $ vanishes identically on the curve $ K(x,y) = 0 $, which holds only for specific symmetric step sets.
• For walks with $ n = 2 $, the sum reduces to $ \frac{y(x_\eta - x)}{z c_\eta} + \frac{y_\xi (x - x_\eta)}{z c(x^2)_\delta} $, and vanishes under the condition $ c/c_\eta = x^2 $, which holds for three symmetric cases.
• For walks with $ n = 3 $ or $ n = 4 $, the sum does not vanish identically on $ K(x,y) = 0 $, explaining why their generating functions are holonomic but not algebraic.
• The kernel curve $ K(x,y) = 0 $ is of genus one for all walks with finite group, enabling uniformization via the Weierstrass $ \wp $-function, which underpins the resolution method.