Random Walks in the Quarter Plane

Abstract

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Asymptotic Lattice Path Enumeration Using Diagonals

“…The fact that this group does not depend on the step set of the model -only on the dimension d -is crucial to obtaining the general results here. When d equals two, the group G matches the group used by [12] and [9]. As we will see in Section 7, G corresponds to the Weyl group of the Weyl chamber A d 1 , where the step set S can be studied in the context of Gessel and Zeilberger [15].…”

“…One key predictor of the nature of a model's generating function (whether it is rational, algebraic, or transcendental D-finite, or none of these) is the order of a group that is associated to each model. This group has its origins in the probabilistic study of random walks, namely [12], and when the group is finite it can sometimes be used to write generating functions as the positive part of an explicit multivariate rational Laurent series. The intimate relation between the generating function of the walks and the nature of the generating function is explored in [7,23].…”

Counting Lattice Walks in the Quarter Plane

“…a south-west jump exists], δ = 0 otherwise. For z = 1/|S|, (1.2) plainly belongs to the generic class of functional equations (arising in the probabilistic context of random walks) studied and solved in the book [4], see section A of our appendix. For general values of z, the analysis of (1.2) for the 79 above-mentioned walks has been carried out in [11], where the integrand of the integral representations is studied in detail, via a complete characterization of ad hoc conformal gluing functions.…”

“…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.…”

“…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]. …”