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Asymptotics of Weighted Reflectable Walks in A₂

Authors Torin Greenwood , Samuel Simon

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Author Details

Torin Greenwood
  • Department of Mathematics, North Dakota State University, Fargo, ND, USA
Samuel Simon
  • Department of Mathematics, Simon Fraser University, Burnaby, Canada

Acknowledgements

This work was started at the 2020-2021 Mathematical Research Community on Combinatorial Applications of Computational Geometry and Algebraic Topology. The authors are grateful for early work with Eric Nathan Stucky and guidance from Marni Mishna.

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Torin Greenwood and Samuel Simon. Asymptotics of Weighted Reflectable Walks in A₂. In 35th International Conference on Probabilistic, Combinatorial and Asymptotic Methods for the Analysis of Algorithms (AofA 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 302, pp. 12:1-12:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024) https://doi.org/10.4230/LIPIcs.AofA.2024.12

Abstract

Lattice walks are used to model various physical phenomena. In particular, walks within Weyl chambers connect directly to representation theory via the Littelmann path model. We derive asymptotics for centrally weighted lattice walks within the Weyl chamber corresponding to A₂ by using tools from analytic combinatorics in several variables (ACSV). We find universality classes depending on the weights of the walks, in line with prior results on the weighted Gouyou-Beauchamps model. Along the way, we identify a type of singularity within a multivariate rational generating function that is not yet covered by the theory of ACSV. We conjecture asymptotics for this type of singularity.

Subject Classification

ACM Subject Classification
  • Mathematics of computing → Generating functions
  • Mathematics of computing → Enumeration
  • Theory of computation → Random walks and Markov chains
Keywords
  • Lattice walks
  • Weyl chambers
  • asymptotics weights
  • analytic combinatorics in several variables

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