Full asymptotic expansion for orbit-summable quadrant walks and discrete polyharmonic functions

TL;DR

This paper develops a method to compute precise asymptotics for the number of quadrant walks with fixed start and end points in orbit-summable models, revealing their connection to discrete polyharmonic functions and extending existing theorems.

Contribution

It introduces a new approach to asymptotic enumeration of quadrant walks using discrete polyharmonic functions, generalizing previous results and accounting for periodicity effects.

Findings

Derived exact asymptotics for orbit-summable quadrant walks.

Connected walk counts to solutions of discrete polyharmonic equations.

Extended a recent theorem on polyharmonic functions and walk enumeration.

Abstract

Enumeration of walks with small steps in the quadrant has been a topic of great interest in combinatorics over the last few years. In this article, it is shown how to compute exact asymptotics of the number of such walks with fixed start- and endpoints for orbit-summable models with finite group, up to arbitrary precision. The resulting representation greatly resembles one conjectured by Chapon, Fusy and Raschel for walks starting from the origin (AofA 2020), differing only in terms appearing due to the periodicity of the model. We will see that the dependency on start- and endpoint is given by discrete polyharmonic functions, which are solutions of $\triangle^n v=0$ for a discretisation $\triangle$ of a Laplace-Beltrami operator. They can be decomposed into a sum of products of lower order polyharmonic functions of either the start- or the endpoint only, which leads to a partial extension of a recent theorem by Denisov and Wachtel (Ann. Prob. 43.3).