Cutoff in the Bernoulli-Laplace Model With Unequal Colors and Urn Sizes

TL;DR

This paper analyzes a generalized Bernoulli-Laplace model with unequal urn sizes and colors, establishing an asymptotic cutoff at log(n) for the mixing time using spectral and coupling techniques.

Contribution

It provides the first rigorous proof of cutoff phenomena in a generalized Bernoulli-Laplace model with unequal urns and colors.

Findings

Mixing time exhibits cutoff at log(n) under certain assumptions.

Spectral analysis and coupling methods are effective for analyzing this model.

Results extend understanding of mixing behaviors in generalized urn models.

Abstract

We consider a generalization of the Bernoulli-Laplace model in which there are two urns and $n$ total balls, of which $r$ are red and $n - r$ white, and where the left urn holds $m$ balls. At each time increment, $k$ balls are chosen uniformly at random from each urn and then swapped. This system can be used to model phenomena such as gas particle interchange between containers or card shuffling. Under a reasonable set of assumptions, we bound the mixing time of the resulting Markov chain asymptotically in $n$ with cutoff at $\log{n}$ and constant window. Among other techniques, we employ the spectral analysis of arXiv:0906.4242 on the Markov transition kernel and the chain coupling tools of arXiv:2203.08647 and arXiv:1606.01437.