Weighted lattice walks and universality classes

Abstract

In this work we consider two different aspects of weighted walks in cones. To begin we examine a particular weighted model, known as the Gouyou-Beauchamps model. Using the theory of analytic combinatorics in several variables we obtain the asymptotic expansion of the total number of Gouyou-Beauchamps walks confined to the quarter plane. Our formulas are parametrized by weights and starting point, and we identify six different asymptotic regimes (called universality classes) which arise according to the values of the weights. The weights allowed in this model satisfy natural algebraic identities permitting an expression of the weighted generating function in terms of the generating function of unweighted walks on the same steps. The second part of this article explains these identities combinatorially for walks in arbitrary cones and dimensions, and provides a characterization of universality classes for general weighted walks. Furthermore, we describe an infinite set of models with non-D-finite generating function.

Constructing discrete harmonic functions in wedges

“…Using the theory of analytic combinatorics in several variables [38], Courtiel et al [10] study drifted simple random walks in various (two-dimensional) wedges and show that the harmonic functions obey to a rigid construction: namely, they are all obtained from a single function (again related to the reflection principle, see [10,Eq. (20)]), after some elementary operations (differentiation, division, evaluation).…”

Walks with Small Steps in the 4D-Orthant

“…Even if we restrict the step sets to subsets of {−1, 0, 1} 2 , sets of so-called small steps, we find that for some step sets the generating function is algebraic; for others, it is not algebraic but still D-finite, and for yet others, it is not even D-finite. This observation sparked an intensive research activity to which many authors have contributed, see [2,4,7,9,12,13,15,20] for some of the milestones and for further references. As a result of this work, the classical setting of walks in the quarter plane is relatively well understood, and the focus of interest is now shifting to the study of variations and generalizations.…”

Higher Dimensional Lattice Walks: Connecting Combinatorial and Analytic Behavior

“…A summary of results is displayed in Table 1. It has previously been observed [22,17] that the subexponential order term n α appearing in asymptotics for the number of walks in a lattice path model has some correlation with the drift of its steps. This phenomenon occurs here: each coordinate where the drift is negative corresponds to a contribution of n −3/2 to dominant asymptotics, a positive drift coordinate does not effect the order term, and a zero drift coordinate corresponds to an asymptotic contribution of at most n −1/2 (depending on whether or not the walk is highly symmetric).…”

“…Although in general one cannot simply determine asymptotics of a model which admits a Hadamard decomposition by multiplying the asymptotics of lower dimensional sequences, many properties such as D-finiteness are inherited via Hadamard decomposition; see Bostan et al [4,Section 5] for details. Given a model whose generating function can be written as a rational diagonal, Courtiel et al [17] develop a method to determine weightings of the step set so that the weighted generating function can be represented as a parametrized rational diagonal with the weights as parameters.…”