Counting Lattice Walks in the Quarter Plane

Abstract

In two recent works [2,1], it has been shown that the counting generating functions (CGF) for the 23 walks with small steps confined in a quarter-plane 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 [4], involving at once algebraic tools and a reduction to boundary value problems. Recently this method has been developed in a combinatorics framework in [11], 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 [4].

Enumeration of three quadrant walks with small steps and walks on other M-quadrant cones

“…In particular, in this section we will show C(x, y; t) is algebraic, D-nite or D-algebraic with respect to x (or y) in the same cases as Q(x, y; t). We note that Fayolle and Raschel also showed that for unweighted models, Q(x, y; t) is D-nite with respect to t in the cases where it is D-nite with respect to x [21], however these results relied on the precise ratios πτ γ that could occur in these cases, so they do not apply so readily to our equation. Nonetheless, we expect that the same result holds for…”

“…In the Section 2, we reduced the problem to nding the unique meromorphic functions A, B : C → C ∪ {∞} and constant F characterised by Theorem 2.8 (for each t). As we discussed, an equation analogous to (19) was found by Raschel for walks in the quarter plane [29], and this equation has since been used to determine precisely when Q(x, y; t) is dierentially algebraic [2,13,22] and to determine in many cases whether it is algebraic or D-nite with respect to x or y [21,23].…”

“…An equation analogous to (19) was found by Raschel for walks in the quarter plane [29], the only dierence being that the transformations z → πτ − γ − z and z → −πτ + γ − z here are z → γ − z and z → −γ − z in the quarter plane. Raschel used this equation to derive an integralexpression solution determining Q(x, y; t) (in the case where the starting point (p, q) = (1, 1)), and the equation has since been used to determine precisely when Q(x, y; t) is dierentially algebraic [2,13,22] and to determine when it is algebraic or D-nite with respect to x or y [21,23].…”

“…The systematic study of walks with small steps in the quarter plane was initiated by Bousquet-Mélou and Mishna in 2010 [7], and since then there has been great progress on the model [3,21,29,26,25,2,23,13,14]. The model is dened as follows: given a step set S ⊂ {−1, 0, 1} 2 \ {(0, 0)}, determine the generating function…”

“…Of the 79 models proposed by Bousquet-Mélou and Mishna, 4 models admit an algebraic generating function [7,3], 19 further models admit a D-nite generating function [7,21], 9 further models admit a D-algebraic generating function [2,23] and the remaining 47 models admit a generating function which is not D-algebraic [13,14], in which case we say that the generating function is D-transcendental. Moreover, in the 74 cases known as non-singular, an exact integral expression is known for the generating function [29], while other exact expressions are known in the 5 singular cases [26,25].…”