PERIODIC POLYA URNS, THE DENSITY METHOD AND ASYMPTOTICS OF YOUNG TABLEAUX

• 成果类型: Article
• 署名作者: Banderier, Cyril; Marchal, Philippe; Wallner, Michael
• 署名单位: Universite Paris 13; Universite Paris 13; Centre National de la Recherche Scientifique (CNRS); Universite de Bordeaux
• 刊物名称: ANNALS OF PROBABILITY
• ISSN/ISSBN: 0091-1798
• DOI: 10.1214/19-AOP1411
• 发表日期: 2020
• 页码: 1921-1965
• 关键词: branching-processes; LIMIT-THEOREMS; maple package; formulas; distributions; walks; sums

摘要

Polya 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 (mod p). We extend the work of Flajolet et al. on Polya 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 Polya 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.