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 focus on the observable given by the total number of balls in the left urn, which converges to a binomial distribution of parameter $1/2$, regardless of the relative number of heavy balls and of the parameter $\alpha$. We study the behavior of this convergence when the total number of balls goes to infinity and show that this can exhibit three different phenomenologies depending on the choice of the two parameters of the model.