On the ergodicity of interacting particle systems under number rigidity

Abstract

In this paper, we provide relations among the following properties: the tail triviality of a probability measure mu on the configuration space {varvec{Upsilon }};the finiteness of a suitable L^2-transportation-type distance bar{textsf {d} }_{varvec{Upsilon }};the irreducibility of local {mu }-symmetric Dirichlet forms on {varvec{Upsilon }}. 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 text {sine}_{2}, text {Airy}_{2}, text {Bessel}_{alpha , 2} (alpha ge 1), and text {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.