# Exposé par Michael Wallner: «Periodic Pólya urns and asymptotics of Young tableaux»

**Speaker:**Michael Wallner

**Location:**BigBlueButton

**Date:**Mer. 10 juin. 2020, 10h30-11h30

La prochaine séance du séminaire Combi du Plateau de Saclay aura lieu mercredi 10 juin à 10h30. Nous aurons le plaisir d’écouter Michael Wallner (LaBRI).

L’exposé se tiendra en ligne par BigBlueButton (lien à demander aux organisateurs).

Le programme du séminaire est disponible ici : https://galac.lri.fr/pages/combi-seminar.html

**Résumé:** *Pólya urns are urns where at each unit of time a ball is drawn uniformly at random and is replaced by 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.*

*Our key tool are generating functions, which encode all possible urn compositions after a certain number of steps. The evolution of the urn is then translated into a system of differential equations and we prove that the moment generating functions are D-finite in one variable. From these we derive asymptotic forms of the moments. 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.*

*Furthermore, we establish some enumerative links with other combinatorial objects yielding a new result on the asymptotics of Young tableaux: We prove that the law of the lower right corner in a triangular Young tableau follows asymptotically a product of generalized Gamma distributions.*

*This is joint work with Cyril Banderier and Philippe Marchal.*