Mixing trichotomy for an Ehrenfest urn with impurities
Matteo Quattropani

TL;DR
This paper analyzes a modified Ehrenfest urn model with two types of balls, revealing how the convergence to equilibrium varies with parameters and identifying three distinct behavioral regimes.
Contribution
It introduces a trichotomy in the asymptotic behavior of the urn model with impurities, depending on the selection rates and heavy ball count.
Findings
Convergence to a binomial distribution with parameter 1/2.
Identification of three different phenomenological regimes.
Analysis of the speed of convergence depending on parameters.
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 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, , 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, 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.
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Taxonomy
TopicsStochastic processes and statistical mechanics · Random Matrices and Applications · Theoretical and Computational Physics
