Cutoff in the Bernoulli-Laplace urn model with swaps of order n

Abstract

This paper considers the $(n,k)$-Bernoulli–Laplace urn model in the case when there are two urns containing n balls each, with two different colors of balls (red and white). In our setting, the total number of red and white balls is the same. Our focus is on the large-time behavior of the corresponding Markov chain tracking the number of red balls in a given urn assuming that the number of selections k at each step obeys $\mathit{\alpha }\le k∕n\le \mathit{\beta }$, where $\mathit{\alpha },\mathit{\beta }$ are constants satisfying $0<\mathit{\alpha }<\mathit{\beta }<\frac{1}{2}$. Under this assumption, cutoff in the total variation distance is established and a cutoff window is provided. The results in this paper solve an open problem posed by Eskenazis and Nestoridi in [8].