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

Joseph S. Alameda; Caroline Bang; Zachary Brennan; David P. Herzog; Jürgen Kritschgau; Elizabeth Sprangel

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arXiv · 2203.08647

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

Joseph S. Alameda·Caroline Bang·Zachary Brennan·David P. Herzog·Jürgen Kritschgau·Elizabeth Sprangel

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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.

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Joseph S. Alameda, Caroline Bang, Zachary Brennan, David P. Herzog, Jürgen Kritschgau, Elizabeth Sprangel. 2022-03-16. Cutoff in the Bernoulli-Laplace model with $O(n)$ swaps. https://arxiv.org/abs/2203.08647

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