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November 2020 Cutoff for the Bernoulli–Laplace urn model with $o(n)$ swaps
Alexandros Eskenazis, Evita Nestoridi
Ann. Inst. H. Poincaré Probab. Statist. 56(4): 2621-2639 (November 2020). DOI: 10.1214/20-AIHP1052


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 $k=1$.

Nous étudions le temps de mélange de l’urne de Bernoulli–Laplace de paramètres $(n,k)$, où $k\in\{0,1,\ldots,n\}$. On considère deux urnes, chacune contenant $n$ boules, telles que combinées elles ont exactement $n$ boules rouges et $n$ boules blanches. A chaque étape du processus, on choisit au hasard $k$ boules dans chaque urne et on les échange. Nous montrons que si $k=o(n)$, le temps de mélange de cette chaîne de Markov exhibe un phénomène de coupure à l’instant $\frac{n}{4k}\log n$ avec une fenêtre d’ordre $\frac{n}{k}\log\log n$. Ceci donne une extension du théorème classique de Diaconis et Shahshahani qui traitait le cas $k=1$.


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Alexandros Eskenazis. Evita Nestoridi. "Cutoff for the Bernoulli–Laplace urn model with $o(n)$ swaps." Ann. Inst. H. Poincaré Probab. Statist. 56 (4) 2621 - 2639, November 2020.


Received: 25 February 2019; Revised: 7 February 2020; Accepted: 23 February 2020; Published: November 2020
First available in Project Euclid: 21 October 2020

MathSciNet: MR4164850
Digital Object Identifier: 10.1214/20-AIHP1052

Primary: 60J10
Secondary: 60C05 , 60G42

Keywords: Bernoulli–Laplace urn model , Cutoff phenomenon , Markov chain , mixing time

Rights: Copyright © 2020 Institut Henri Poincaré

Vol.56 • No. 4 • November 2020
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