Cutoff for generalised Bernoulli-Laplace urn models

TL;DR

This paper introduces a multi-colour, multi-urn generalization of the Bernoulli-Laplace model, analyzing its mixing time and cutoff phenomena as the number of balls grows large, with applications to card shuffling.

Contribution

It extends the Bernoulli-Laplace urn model to multiple colours and urns, establishing cutoff conditions for the mixing time in this generalized setting.

Findings

Cutoff occurs when the single-ball chain on [d] is irreducible.

The cutoff phenomenon also applies to a labeled version of the model.

Partial results are obtained for a card shuffling variant of the model.

Abstract

We introduce a multi-colour multi-urn generalisation of the Bernoulli-Laplace urn model, consisting of $d$ urns, $m$ colours, and $dmn$ balls, with $dn$ balls of each colour and $mn$ balls in each urn. At each step, one ball is drawn uniformly at random from each urn, and the chosen balls are redistributed among the urns based on a permutation drawn from a distribution $\mu$ on the symmetric group $S_d$. We study the mixing time of this Markov chain for fixed $m$, $d$, and $\mu$, as $n \rightarrow \infty$. We show that there is cutoff whenever the chain on $[d]$ corresponding to the evolution of a single ball is irreducible, and that the same holds for a labeled version of the model. As an application, we also obtain partial results on cutoff for a card shuffling version of the model in which the cards are labeled and their ordering within each stack matters.