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arXiv · 2401.00837

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

Alexander Kroitor·Stephen Melczer
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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.

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Alexander Kroitor, Stephen Melczer. 2024-01-01. Completing the Asymptotic Classification of Mostly Symmetric Short Step Walks in an Orthant. https://arxiv.org/abs/2401.00837

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