Cutoff for the Bernoulli-Laplace urn model with $o(n)$ swaps

TL;DR

This paper analyzes the mixing time cutoff for the Bernoulli-Laplace urn model with o(n) swaps, extending classical results to a broader range of swap sizes and providing precise cutoff and window estimates.

Contribution

It extends the classical cutoff result of Diaconis and Shahshahani to cases where the number of swaps k is o(n), establishing the cutoff time and window for these scenarios.

Findings

Mixing time cutoff occurs at (n/4k) log n for k=o(n).

Cutoff window is of order (n/k) log log n.

Results generalize classical case k=1 to larger swap sizes.

Abstract

We study the mixing time of the $(n,k)$ Bernoulli--Laplace urn model, where $k\in\{0,1,\ldots,n\}$. Consider two urns, each containing $n$ balls, so that when combined they have precisely $n$ red balls and $n$ white balls. At each step of the process choose uniformly at random $k$ balls from the left urn and $k$ balls from the right urn and switch them simultaneously. We show that if $k=o(n)$, this Markov chain exhibits mixing time cutoff at $\frac{n}{4k}\log n$ and window of the order $\frac{n}{k}\log\log n$. This is an extension of a classical theorem of Diaconis and Shahshahani who treated the case $k=1$.