Mixing trichotomy for an Ehrenfest urn with impurities

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 $\alpha \in (0,1)$ 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 $m_N\in \{1,\dots ,N-1\}$. We focus on the observable given by the total number of balls in the left urn, which converges to a binomial distribution of parameter $1\mathrm{/}2$, regardless of the choice of the two parameters, $\alpha$ and $m_N$. 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.