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arXiv · 2111.05027

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

Rodolphe Garbit·Kilian Raschel
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Abstract

A. 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($\tau$ > n), where $\tau$ is the first exit time from a given cone; the second series is that of the excursion probabilities P($\tau$ > 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 wellknown 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, elementary and selfcontained proof that it cannot be rational neither.

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Rodolphe Garbit, Kilian Raschel. 2021-11-09. The generating function of the survival probabilities in a cone is not rational. https://arxiv.org/abs/2111.05027

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