We consider exclusion processes on a rooted $d$-regular tree. We start from a Bernoulli product measure conditioned on having a particle at the root, which we call the tagged particle. For $d\geq 3$, we show that the tagged particle has positive linear speed and satisfies a central limit theorem. We give an explicit formula for the speed. As a key step in the proof, we first show that the exclusion process “seen from the tagged particle” has an ergodic invariant measure.
"Limit theorems for the tagged particle in exclusion processes on regular trees." Electron. Commun. Probab. 24 1 - 10, 2019. https://doi.org/10.1214/18-ECP205