Non-D-Finite Walks in a Three-Quadrant Cone

  • Sami MustaphaEmail author


We prove that the sequence \(\left( e_n^{\Gamma }\right) _{n\in \mathbb {N}}\) of numbers of excursions in a three-quadrant cone corresponding to a non-singular step set \(\Gamma \subset \{0, \pm 1\}^2\) with an infinite group does not satisfy any nontrivial linear recurrence with polynomial coefficients. This gives a positive answer to a question asked by M. Bousquet-Mélou about D-finiteness of the trivariate generating function of the numbers of walks with given length and prescribed ending point. In the process, we determine the asymptotics of \(e_n^{\Gamma }\), \(n\rightarrow \infty \), for the 74 non-singular two-dimensional models, giving the first complete computation of excursions asymptotics in a non-convex cone. Moreover, using potential theoretic comparison arguments, we give the asymptotics of the number of walks avoiding the negative quadrant and of length n for all non-singular step sets having zero drift.


Lattice path enumeration Generating functions D-finiteness Random walks Discrete potential theory 

Mathematics Subject Classification

05A16 60G50 



I would like to thank M. Bousquet-Mélou who asked me a question, during a talk I gave at Université de Bordeaux, about a 5 / 3 exponent that appears in the three-quadrant theory, and with whom I had stimulating exchanges on the subject. I also wish to thank an anonymous referee for valuable comments and suggestions.


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Authors and Affiliations

  1. 1.Institut de Mathématiques de Jussieu-Paris Rive GaucheSorbonne UniversitéParis Cedex 05France

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