Open Access
July 2020 Periodic Pólya urns, the density method and asymptotics of Young tableaux
Cyril Banderier, Philippe Marchal, Michael Wallner
Ann. Probab. 48(4): 1921-1965 (July 2020). DOI: 10.1214/19-AOP1411


Pólya urns are urns where at each unit of time a ball is drawn and replaced with some other balls according to its colour. We introduce a more general model: the replacement rule depends on the colour of the drawn ball and the value of the time ($\operatorname{mod}p$). We extend the work of Flajolet et al. on Pólya urns: the generating function encoding the evolution of the urn is studied by methods of analytic combinatorics. We show that the initial partial differential equations lead to ordinary linear differential equations which are related to hypergeometric functions (giving the exact state of the urns at time $n$). When the time goes to infinity, we prove that these periodic Pólya urns have asymptotic fluctuations which are described by a product of generalized gamma distributions. With the additional help of what we call the density method (a method which offers access to enumeration and random generation of poset structures), we prove that the law of the southeast corner of a triangular Young tableau follows asymptotically a product of generalized gamma distributions. This allows us to tackle some questions related to the continuous limit of random Young tableaux and links with random surfaces.


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Cyril Banderier. Philippe Marchal. Michael Wallner. "Periodic Pólya urns, the density method and asymptotics of Young tableaux." Ann. Probab. 48 (4) 1921 - 1965, July 2020.


Received: 1 March 2019; Revised: 1 October 2019; Published: July 2020
First available in Project Euclid: 20 July 2020

zbMATH: 07224964
MathSciNet: MR4124529
Digital Object Identifier: 10.1214/19-AOP1411

Primary: 60C05
Secondary: 05A15 , 60F05 , 60K99

Keywords: analytic combinatorics , D-finite function , generalized gamma distribution , generating functions , hypergeometric function , Mittag-Leffler distribution , Pólya urn , pumping moment , Young tableau

Rights: Copyright © 2020 Institute of Mathematical Statistics

Vol.48 • No. 4 • July 2020
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