Mixing trichotomy for an Ehrenfest urn with impurities

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 $1$ 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 $1/2$, 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.