On the ergodicity of interacting particle systems under number rigidity

In this paper, we provide relations among the following properties: (a) the tail triviality of a probability measure mu on the configuration space Gamma; (b) the finiteness of a suitable L-2-transportation-type distance (d) over bar (Gamma); (c) the irreducibility of local mu-symmetric Dirichlet forms on Gamma. As an application, we obtain the ergodicity (i.e., the convergence to the equilibrium) of interacting infinite diffusions having logarithmic interaction and arising from determinantal/permanental point processes including sine(2), Airy(2), Bessel(alpha,2) (alpha >= 1), and Ginibre point processes. In particular, the case of the unlabelled Dyson Brownian motion is covered. For the proof, the number rigidity of point processes in the sense of Ghosh-Peres plays a key role.