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Periodic Pólya Urns and an Application to Young Tableaux

Authors Cyril Banderier , Philippe Marchal , Michael Wallner

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Author Details

Cyril Banderier
  • Université Paris 13, LIPN, UMR CNRS 7030, http://lipn.univ-paris13.fr/~banderier
Philippe Marchal
  • Université Paris 13, LAGA, UMR CNRS 7539, https://math.univ-paris13.fr/~marchal
Michael Wallner
  • Université de Bordeaux, LaBRI, UMR CNRS 5800, http://dmg.tuwien.ac.at/mwallner

Funding

This work was initiated during the postdoctoral position of Michael Wallner at the University of Paris Nord, in September-December 2017, thanks to a MathStic funding. The subsequent part of this collaboration was funded by the Erwin Schrödinger Fellowship of the Austrian Science Fund (FWF): J 4162-N35.

Cite AsGet BibTex

Cyril Banderier, Philippe Marchal, and Michael Wallner. Periodic Pólya Urns and an Application to Young Tableaux. In 29th International Conference on Probabilistic, Combinatorial and Asymptotic Methods for the Analysis of Algorithms (AofA 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 110, pp. 11:1-11:13, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)
https://doi.org/10.4230/LIPIcs.AofA.2018.11

Abstract

Pólya urns are urns where at each unit of time a ball is drawn and is 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 (mod p). We discuss some intriguing properties of the differential operators associated to the generating functions encoding the evolution of these urns. The initial non-linear partial differential equation indeed leads to linear differential equations and we prove that the moment generating functions are D-finite. For a subclass, we exhibit a closed form for the corresponding generating functions (giving the exact state of the urns at time n). When the time goes to infinity, we show that these periodic Pólya urns follow a rich variety of behaviours: their asymptotic fluctuations are described by a family of distributions, the generalized Gamma distributions, which can also be seen as powers of Gamma distributions. En passant, we establish some enumerative links with other combinatorial objects, and we give an application for a new result on the asymptotics of Young tableaux: This approach allows us to prove that the law of the lower right corner in a triangular Young tableau follows asymptotically a product of generalized Gamma distributions.

Subject Classification

ACM Subject Classification
  • Mathematics of computing → Enumeration
  • Mathematics of computing → Generating functions
  • Mathematics of computing → Distribution functions
  • Mathematics of computing → Ordinary differential equations
  • Theory of computation → Generating random combinatorial structures
  • Theory of computation → Random walks and Markov chains
Keywords
  • Pólya urn
  • Young tableau
  • generating functions
  • analytic combinatorics
  • pumping moment
  • D-finite function
  • hypergeometric function
  • generalized Gamma distribution
  • Mittag-Leffler distribution

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