A rapidly mixing stochastic system of finite interacting particles on the circle

We analyze the speed of convergence to stationarity for a specific stochastic system consisting of a finite number of interacting particles on the circle. We define a coupling and a martingale related to this coupling to show that the time needed to approach stationarity is a polynomial in the number of particles of degree at most 12, and thus prove that the chain is rapidly mixing. This is partly due to the fact that the coupling time happens before the martingale escapes from a certain strip. We use a relaxation time related to Poincaré's characterization of the second largest eigenvalue of the chain, to lower bound the time to stationarity by a polynomial of degree 3.