Séminaire Lotharingien de Combinatoire, 87B.11 (2023), 17 pp.

Rodolphe Garbit and Kilian Raschel

The generating function of the survival probabilities in a cone is not rational

Abstract. We look at multidimensional random walks (Sn)n≥0 in convex cones, and address the question of whether two naturally associated generating functions may define rational functions. The first series is the one of the survival probabilities P(τ>n), where τ is the first exit time from a given cone; the second series is that of the excursion probabilities P(τ>n,Sn=y). Our motivation to consider this question is twofold: first, it goes along with a global effort of the combinatorial community to classify the algebraic nature of the series counting random walks in cones; second, rationality questions of the generating functions are strongly associated with the asymptotic behaviors of the above probabilities, which have their own interest. Using well-known relations between rationality of a series and possible asymptotics of its coefficients, recent probabilistic estimates immediately imply that the excursion generating function is not rational. Regarding the survival probabilities generating function, we propose a short and self-contained proof that it cannot be rational neither.

Received: November 23, 2021. Revised: August 06, 2022. Published: August 31, 2023.

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If you need to cite this article, here is a bibtex:

@article{GarbitRaschel23,
author = {Garbit, Rodolphe and Raschel, Kilian},
title = {The generating function of the survival probabilities in a cone is not rational},
fjournal = {S{\'e}minaire Lotharingien de Combinatoire},
journal = {S{\'e}min. Lothar. Combin.},
issn={1286-4889},
volume = {87B},
number= {11},
pages = {1--17},
year = {2023},
language = {English},
doi={10.48550/arXiv.2111.05027},
}