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# Periodic Pólya Urns and an Application to Young Tableaux

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LIPIcs.AofA.2018.11.pdf
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## Cite As

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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## References

1. Omer Angel, Alexander E. Holroyd, Dan Romik, and Bálint Virág. Random sorting networks. Adv. Math., 215(2):839-868, 2007. URL: http://dx.doi.org/10.1016/j.aim.2007.05.019.
2. Krishna B. Athreya and Peter E. Ney. Branching processes. Springer, 1972.
3. Amitava Bagchi and Asim K. Pal. Asymptotic normality in the generalized Pólya-Eggenberger urn model, with an application to computer data structures. SIAM J. Algebraic Discrete Methods, 6(3):394-405, 1985. URL: http://dx.doi.org/10.1137/0606041.
4. Jean Bertoin and Marc Yor. On subordinators, self-similar Markov processes and some factorizations of the exponential variable. Electron. Comm. Probab., 6:95-106, 2001. URL: https://projecteuclid.org/euclid.ecp/1461097555.
5. Torsten Carleman. Sur les équations intégrales singulières à noyau réel et symétrique. Uppsala Universitets Årsskrift, 1923.
6. Brigitte Chauvin, Cécile Mailler, and Nicolas Pouyanne. Smoothing equations for large Pólya urns. J. Theoret. Probab., 28(3):923-957, 2015. URL: http://dx.doi.org/10.1007/s10959-013-0530-z.
7. Florian Eggenberger and George Pólya. Über die Statistik verketteter Vorgänge. Z. Angew. Math. Mech., 3:279-290, 1923. URL: http://dx.doi.org/10.1002/zamm.19230030407.
8. Florian Eggenberger and George Pólya. Sur l'interprétation de certaines courbes de fréquence. C. R. Acad. Sc., 187:870-872, 1928. URL: http://visualiseur.bnf.fr/CadresFenetre?O=NUMM-3140&I=2&M=tdm.
9. Giulia Fanti and Pramod Viswanath. Deanonymization in the Bitcoin P2P Network. Proceedings of the 31st Conference on Neural Information Processing Systems, 2017. URL: https://papers.nips.cc/paper/6735-deanonymization-in-the-bitcoin-p2p-network.pdf.
10. Philippe Flajolet, Philippe Dumas, and Vincent Puyhaubert. Some exactly solvable models of urn process theory. Discrete Math. Theor. Comput. Sci. Proc., AG:59-118, 2006. Fourth Colloquium on Mathematics and Computer Science Algorithms, Trees, Combinatorics and Probabilities. URL: https://hal.inria.fr/hal-01184710/document.
11. Philippe Flajolet, Joaquim Gabarró, and Helmut Pekari. Analytic urns. Ann. Probab., 33(3):1200-1233, 2005. URL: http://dx.doi.org/10.1214/009117905000000026.
12. Philippe Flajolet and Robert Sedgewick. Analytic Combinatorics. Cambridge University Press, 2009. URL: http://algo.inria.fr/flajolet/Publications/book.pdf.
13. Maurice Fréchet and James Shohat. A proof of the generalized second-limit theorem in the theory of probability. Trans. Amer. Math. Soc., 33(2):533-543, 1931. URL: http://dx.doi.org/10.2307/1989421.
14. Curtis Greene, Albert Nijenhuis, and Herbert S. Wilf. Another probabilistic method in the theory of Young tableaux. J. Comb. Theory, Ser. A, 37:127-135, 1984. URL: http://dx.doi.org/10.1016/0097-3165(84)90065-7.
15. Svante Janson. Functional limit theorems for multitype branching processes and generalized Pólya urns. Stochastic Process. Appl., 110(2):177-245, 2004. URL: http://dx.doi.org/10.1016/j.spa.2003.12.002.
16. Svante Janson. Asymptotic degree distribution in random recursive trees. Random Structures Algorithms, 26(1-2):69-83, 2005. URL: http://dx.doi.org/10.1002/rsa.20046.
17. Svante Janson. Moments of Gamma type and the Brownian supremum process area. Probab. Surv., 7:1-52, 2010. With addendum on pages 207-208. URL: http://dx.doi.org/10.1214/10-PS160.
18. Richard Kenyon. Dominos and the Gaussian free field. Ann. Probab., 29(3):1128-1137, 2001. URL: http://dx.doi.org/10.1214/aop/1015345599.
19. Morteza Khodabin and Alireza Ahmadabadi. Some properties of generalized gamma distribution. Mathematical Sciences Quarterly Journal, 4(1):9-27, 2010. URL: http://mathscience.kiau.ac.ir/Content/Vol4No1/2.pdf.
20. Markus Kuba and Henning Sulzbach. On martingale tail sums in affine two-color urn models with multiple drawings. J. Appl. Probab., 54(1):96-117, 2017. URL: http://dx.doi.org/10.1017/jpr.2016.89.
21. Ian G. Macdonald. Symmetric functions and Hall polynomials. Oxford Classic Texts in the Physical Sciences. The Clarendon Press, Oxford University Press, second edition, 2015.
22. Hosam M. Mahmoud. Pólya urn models. CRC Press, 2009.
23. Cécile Mailler. Describing the asymptotic behaviour of multicolour Pólya urns via smoothing systems analysis. arXiv, 2014. URL: https://arxiv.org/abs/1407.2879.
24. Philippe Marchal. Rectangular Young tableaux and the Jacobi ensemble. Discrete Math. Theor. Comput. Sci. Proceedings, BC:839-850, 2016. URL: http://www.lix.polytechnique.fr/~pilaud/FPSAC16/final_3.
25. Erol A. Peköz, Adrian Röllin, and Nathan Ross. Generalized gamma approximation with rates for urns, walks and trees. Ann. Probab., 44(3):1776-1816, 2016. URL: http://dx.doi.org/10.1214/15-AOP1010.
26. Boris Pittel and Dan Romik. Limit shapes for random square Young tableaux. Adv. Appl. Math., 38(2):164-209, 2007. URL: http://dx.doi.org/10.1016/j.aam.2005.12.005.
27. George Pólya. Sur quelques points de la théorie des probabilités. Annales de l'Institut Henri Poincaré, 1:117-161, 1930. URL: http://www.numdam.org/article/AIHP_1930__1_2_117_0.pdf.
28. Ronald L. Rivest. Tabulation audits: Explained and extended. arXiv, 2018. URL: https://arxiv.org/abs/1801.00528.
29. Dan Romik. The surprising mathematics of longest increasing subsequences. Institute of Mathematical Statistics Textbooks. Cambridge University Press, New York, 2015. URL: https://www.math.ucdavis.edu/~romik/download-book.php.
30. Robert T. Smythe and Hosam M. Mahmoud. A survey of recursive trees. Theory Probab. Math. Statist., 51:1-29, 1994.
31. Edney W. Stacy. A generalization of the gamma distribution. Ann. Math. Statist., 33:1187-1192, 1962. URL: http://dx.doi.org/10.1214/aoms/1177704481.
32. Anatoly M. Vershik. Randomization of algebra and algebraization of probability. In Mathematics unlimited - 2001 and beyond., pages 1157-1166. Springer, 2001.
33. Hubert Stanley Wall. Analytic Theory of Continued Fractions. D. Van Nostrand Company, Inc., New York, 1948. (See also the reprint by Chelsea in 1967.).