We consider a nonreversible finite Markov chain called Repeated Balls-into-Bins (RBB) process. This process is a discrete time conservative interacting particle system with parallel updates. Place initially in $L$ bins $rL$ balls, where $r$ is a fixed positive constant. At each time step a ball is removed from each non-empty bin. Then all these removed balls are uniformly reassigned into bins. We prove that the mixing time of the RBB process is of order $L$. Furthermore we show that if the initial configuration has $o(L)$ balls per site the equilibrium is attained in $o(L)$ steps.
"Mixing time for the Repeated Balls into Bins dynamics." Electron. Commun. Probab. 25 1 - 14, 2020. https://doi.org/10.1214/20-ECP338