Completing the Asymptotic Classification of Mostly Symmetric Short Step Walks in an Orthant

Abstract

In recent years, the techniques of analytic combinatorics in several variables (ACSV) have been applied to determine asymptotics for several families of lattice path models restricted to the orthant $\mathbb{N}^d$ and defined by step sets $\mathcal{S}\subset\{-1,0,1\}^d\setminus\{\mathbf{0}\}$. Using the theory of ACSV for smooth singular sets, Melczer and Mishna determined asymptotics for the number of walks in any model whose set of steps $\mathcal{S}$ is "highly symmetric" (symmetric over every axis). Building on this work, Melczer and Wilson determined asymptotics for all models where $\mathcal{S}$ is "mostly symmetric" (symmetric over all but one axis) *except* for models whose set of steps have a vector sum of zero but are not highly symmetric. In this paper we complete the asymptotic classification of the mostly symmetric case by analyzing a family of saddle-point-like integrals whose amplitudes are singular near their saddle points.