Open Access
2024 Cutoff in the Bernoulli-Laplace urn model with swaps of order n
Joseph S. Alameda, Caroline Bang, Zachary Brennan, David P. Herzog, Jürgen Kritschgau, Elizabeth Sprangel
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Electron. Commun. Probab. 29: 1-13 (2024). DOI: 10.1214/23-ECP569


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 αknβ, where α,β are constants satisfying 0<α<β<12. 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].


We graciously acknowledge support from the National Science Foundation through grants DMS-1855504 (D.P.H.) and DMS-1839918 (C.B., Z.B., J.K., E.S.). We also acknowledge fruitful conversations with Evita Nestoridi and Julia Komjathy about the topics of this paper. We thank the anonymous referees for helpful comments and suggestions, and for helping improve the strength of our main result from an earlier version of the paper.


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Joseph S. Alameda. Caroline Bang. Zachary Brennan. David P. Herzog. Jürgen Kritschgau. Elizabeth Sprangel. "Cutoff in the Bernoulli-Laplace urn model with swaps of order n." Electron. Commun. Probab. 29 1 - 13, 2024.


Received: 15 August 2022; Accepted: 15 December 2023; Published: 2024
First available in Project Euclid: 9 January 2024

Digital Object Identifier: 10.1214/23-ECP569

Primary: 37A25 , 60J10

Keywords: Bernoulli-Laplace model , Cutoff , Markov chain , Mixing times

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