Higher Dimensional Lattice Walks: Connecting Combinatorial and Analytic Behavior

TL;DR

This paper advances the analysis of lattice walks in higher dimensions by deriving asymptotics for models with complex singularities, extending previous methods, and confirming conjectured asymptotics for specific two-dimensional cases.

Contribution

It develops new asymptotic formulas for lattice walks with non-smooth singularities, connecting combinatorial and analytic techniques in higher dimensions.

Findings

Asymptotics for models with asymmetric step sets

Closed-form asymptotics for certain symmetric models

Rigorous proof of conjectured 2D walk asymptotics

Abstract

We consider the enumeration of walks on the non-negative lattice $\mathbb{N}^d$, with steps defined by a set $\mathcal{S} \subset \{-1, 0, 1\}^d \setminus \{\mathbf{0}\}$. 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 $\mathcal{S}$ is symmetric over every axis; in this setting one can always apply the methods of ACSV to a multivariate rational function whose 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 with 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 which 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.