On the Ergodicity of Interacting Particle Systems under Number Rigidity

Abstract

In this paper, we provide relations among the following properties: (a) the tail triviality of a probability measure $μ$ on the configuration space ${\boldsymbolΥ}$; (b) the finiteness of the $L^2$-transportation-type distance $\bar{\mathsf d}_{\boldsymbolΥ}$; (c) the irreducibility of $μ$-symmetric Dirichlet forms on ${\boldsymbolΥ}$. As an application, we obtain the ergodicity (i.e., the convergence to the equilibrium) of interacting infinite diffusions having logarithmic interaction arisen from determinantal/permanental point processes including $\mathrm{sine}_{2}$, $\mathrm{Airy}_{2}$, $\mathrm{Bessel}_{α, 2}$ ($α\ge 1$), and $\mathrm{Ginibre}$ point processes, in particular, the case of 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.