Abstract
We consider a version of the classical Ehrenfest urn model with two urns and two types of balls: regular and heavy. Each ball is selected independently according to a Poisson process having rate 1 for regular balls and rate for heavy balls, and once a ball is selected, is placed in a urn uniformly at random. We study the asymptotic behavior when the total number of balls, N, goes to infinity, and the number of heavy ball is set to . We focus on the observable given by the total number of balls in the left urn, which converges to a binomial distribution of parameter , regardless of the choice of the two parameters, α and . We study the speed of convergence and show that this can exhibit three different phenomenologies depending on the choice of the two parameters of the model.
Acknowledgments
The author is a member of GNAMPA-INdAM, and he thanks the German Research Foundation (project number 444084038, priority program SPP2265) for financial support. Moreover, the author wishes to thank Pietro Caputo and Federico Sau for helpful discussions on the topic, and the anonymous referee for having suggested several key improvements on the first draft of this paper.
Citation
Matteo Quattropani. "Mixing trichotomy for an Ehrenfest urn with impurities." Electron. Commun. Probab. 29 1 - 13, 2024. https://doi.org/10.1214/24-ECP610
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