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On the Nature of Four Models of Symmetric Walks Avoiding a Quadrant

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Abstract

We study the nature of the generating series of some models of walks with small steps in the three quarter plane. More precisely, we restrict ourselves to the situation where the group is infinite, the kernel has genus one, and the step set is diagonally symmetric (i.e., with no steps in anti-diagonal directions). In that situation, after a transformation of the plane, we derive a quadrant-like functional equation. Among the four models of walks, we obtain, using difference Galois theory, that three of them have a differentially transcendental generating series, and one has a differentially algebraic generating series.

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Notes

  1. Thereafter, the expressions avoiding a quadrant, confined to the three quadrants, and confined to the three quarter plane, will be used with no distinction.

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Acknowledgements

The authors want to warmly thank the referees for their detailed and helpful comments. The authors also thank Samuel Simon for his help with the English language, Kilian Raschel, and Manuel Kauers, for their suggestions.

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Authors and Affiliations

  1. Institut de Recherche Mathématique Avancée, U.M.R. 7501 Université de Strasbourg et C.N.R.S., 7, rue René Descartes, 67084, Strasbourg, France

    Thomas Dreyfus

  2. Institute for Algebra, Johannes Kepler University, Altenbergerstrasse, 69 4040, Linz, Austria

    Amélie Trotignon

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  1. Thomas Dreyfus
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  2. Amélie Trotignon
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Correspondence to Thomas Dreyfus.

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Communicated by Marni Mishna.

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This project has received funding from the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme under the Grant Agreement No. 759702. This project has received funding from the ANR de rerum natura ANR-19-CE40-0018. A. Trotignon was supported by the Austrian Science Fund (FWF) grant FWF05004.

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Dreyfus, T., Trotignon, A. On the Nature of Four Models of Symmetric Walks Avoiding a Quadrant. Ann. Comb. 25, 617–644 (2021). https://doi.org/10.1007/s00026-021-00541-8

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  • DOI: https://doi.org/10.1007/s00026-021-00541-8

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