Periodic P\'olya Urns, the Density Method, and Asymptotics of Young Tableaux

TL;DR

This paper generalizes Pólya urn models with periodic replacement rules, analyzes their asymptotic behavior using analytic combinatorics, and applies these results to the study of Young tableaux and related combinatorial structures.

Contribution

It extends Pólya urn theory to periodic cases, derives differential equations linked to hypergeometric functions, and connects these to the asymptotic distribution of Young tableaux corners using the density method.

Findings

Periodic Pólya urns have asymptotic fluctuations described by generalized gamma distributions.

The law of the south-east corner of a Young tableau converges to a product of generalized gamma distributions.

The density method enables analysis of large random Young tableaux and their continuous limits.

Abstract

P\'olya 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 ($\operatorname{mod} p$). We extend the work of Flajolet et al. on P\'olya 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 P\'olya 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 south-east 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 large random Young tableaux and links with random surfaces.