Cutoff in the Bernoulli-Laplace model with $O(n)$ swaps

TL;DR

This paper analyzes the mixing time of the Bernoulli-Laplace model with a number of swaps proportional to the total number of balls, establishing cutoff phenomena and partially resolving an open problem.

Contribution

It proves cutoff in the Bernoulli-Laplace model for $O(n)$ swaps and provides a cutoff window, addressing an open problem in the field.

Findings

Cutoff occurs in the total variation distance for the model.

A cutoff window is explicitly characterized.

Results partially resolve a previously posed open problem.

Abstract

This paper considers the $(n,k)$-Bernoulli--Laplace model in the case when there are two urns, the total number of red and white balls is the same, and the number of selections $k$ at each step is on the same asymptotic order as the number of balls $n$ in each urn. Our main focus is on the large-time behavior of the corresponding Markov chain tracking the number of red balls in a given urn. Under reasonable assumptions on the asymptotic behavior of the ratio $k/n$ as $n\rightarrow \infty$, cutoff in the total variation distance is established. A cutoff window is also provided. These results, in particular, partially resolve an open problem posed by Eskenazis and Nestoridi.