2019/12/02 by Cyril Banderier, Banderier, Cyril, Philippe Marchal +3
Mathematics · #05A15 #60C05 #60F05 #60K99 #Advanced Combinatorial Mathematics #Combinatorics (math.CO) #Discrete Mathematics (cs.DM) #FOS: Computer and information sciences #FOS: Mathematics #Probability (math.PR) #Random Matrices and Applications #Stochastic processes and statistical mechanics
paper · pdf · doi:10.48550/arxiv.1912.01035
openalex publication_date 2019/12/02 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
P 'olya urns are urns where at each unit of time a ball is drawn and replaced\nwith some other balls according to its colour. We introduce a more general\nmodel: the replacement rule depends on the colour of the drawn ball and the\nvalue of the time (\mod p). We extend the work of Flajolet et\nal. on P 'olya urns: the generating function encoding the evolution of the urn\nis studied by methods of analytic combinatorics. We show that the initial\npartial differential equations lead to ordinary linear differential equations\nwhich are related to hypergeometric functions (giving the exact state of the\nurns at time n). When the time goes to infinity, we prove that these periodic\nP 'olya urns have asymptotic fluctuations which are described by a product of\ngeneralized gamma distributions. With the additional help of what we call the\ndensity method (a method which offers access to enumeration and random\ngeneration of poset structures), we prove that the law of the south-east corner\nof a triangular Young tableau follows asymptotically a product of generalized\ngamma distributions. This allows us to tackle some questions related to the\ncontinuous limit of large random Young tableaux and links with random surfaces.\n