Higher Dimensional Lattice Walks: Connecting Combinatorial and Analytic Behavior

Abstract

We consider the enumeration of walks on the non-negative lattice N d , with steps defined by a set S ⊂ {−1, 0, 1} d \{0}. Previous work in this area has established asymptotics for the number of walks in certain families of models by applying the techniques of analytic combinatorics in several variables (ACSV), where one encodes the generating function of a lattice path model as the diagonal of a multivariate rational function. Melczer and Mishna obtained asymptotics when the set of steps S is symmetric over every axis; in this setting one can always apply the methods of ACSV to a multivariate rational function whose set of singularities is a smooth manifold (the simplest case). Here we go further, providing asymptotics for models with generating functions that must be encoded by multivariate rational functions having non-smooth singular sets. In the process, our analysis connects past work to deeper structural results in the theory of analytic combinatorics in several variables. One application is a closed form for asymptotics of models defined by step sets that are symmetric over all but one axis. As a special case, we apply our results when d = 2 to give a rigorous proof of asymptotics conjectured by Bostan and Kauers; asymptotics for walks returning to boundary axes and the origin are also given.

Polyharmonic functions and random processes in cones

“…Suppose now that v is bi-harmonic and satisfies Gv = h, where h is harmonic. The functional equation for v now reads (19) γ(x, y)…”

“…The same computation applies to L 2 (v). As such, using the equation (19), the Laplace transform of the bi-harmonic function v admits the following form:…”

“…As in the continuous setting, we first investigate the appearance of polyharmonic functions in asymptotic expansions of coefficients counting walks with fixed endpoints in the quarter plane. Several techniques may apply to obtain asymptotic expansions starting from exact expression for enumeration of lattice paths, e.g., Laplace's method and saddle point method [17] or analytic combinatorics in several variables [19]. We will present such an asymptotic expansion in the simplest case of the diagonal walk via Laplace's method.…”

“…• We shine a light on a new link between discrete polyharmonic functions and complete asymptotic expansions in the enumeration of walks. • We introduce a new class of functional equations (see (19) and ( 26)) for which the method of Tutte's invariants introduced in [22,5,6] proves to be useful. • In the unweighted planar case, it has been shown [8] that knowing the rationality of the exponent α 0 in (7) was sufficient to decide the non-D-finiteness of the series of excursions.…”