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

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