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

Alexander Kroitor, Stephen Melczer

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.

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 and defined by step sets . 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 is "highly symmetric" (symmetric over every axis). Building on this work, Melczer and Wilson determined asymptotics for all models where 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.
Paper Structure (14 sections, 11 theorems, 97 equations, 1 table)

This paper contains 14 sections, 11 theorems, 97 equations, 1 table.

Table of Contents

  1. Models with Highly and Mostly Symmetric Step Sets
  2. Our Results
  3. ACSV and Lattice Path Enumeration
  4. Highly Symmetric Models
  5. Mostly Symmetric Models
  6. Negative Drift Models
  7. Positive Drift Models
  8. Zero Drift Models
  9. Asymptotics of Zero Drift Mostly Symmetric Models
  10. Residue Computations
  11. Running assumptions on $\epsilon$ and $\delta$
  12. Computing Dominant Asymptotics
  13. Conclusion
  14. Acknowledgements

Key Result

Theorem 2

Let $\mathcal{S} \subset \{-1,0,1\}^d \setminus \{\mathbf{0}\}$ be a set of steps that is symmetric over every axis and moves forwards and backwards in each coordinate. Then the total weight $s_n$ of walks of length $n$ taking steps in $\mathcal{S}$, beginning at the origin, and never leaving $\math

Theorems & Definitions (28)

  • Remark 1
  • Theorem 2: Highly Symmetric Asymptotics MelczerMishna2016
  • Theorem 3: Positive Drift Asymptotics MelczerWilson2019
  • Theorem 4: Negative Drift Asymptotics MelczerWilson2019
  • Remark 5
  • Theorem 6
  • Remark 7
  • Remark 8
  • Example 9
  • Example 10
  • ...and 18 more