On the ergodicity of interacting particle systems under number rigidity

Abstract

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