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We consider a version of the classical Ehrenfest urn model with two urns andtwo types of balls: regular and heavy. Each ball is selected independentlyaccording to a Poisson process having rate $1$ for regular balls and rate$\alpha\in(0,1)$ for heavy balls, and once a ball it is selected is placed in aurn uniformly at random. We study the asymptotic behavior when the total numberof balls, $N$, goes to infinity, and the number of heavy ball is set to$\lfloor N^\beta\rfloor$ for some $\beta\in[0,1]$. We focus on the observablegiven by the total number of balls in the left urn, which converges to abinomial distribution of parameter $1/2$, regardless of the choice of the twoparameters, $\alpha$ and $\beta$. We study the speed of convergence and showthat this can exhibit three different phenomenologies depending on the choiceof the two parameters of the model.

Mixing trichotomy for an Ehrenfest urn with impurities