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 $\mu$ on the configuration space ${\boldsymbol\Upsilon}$; (b) the finiteness of the $L^2$-transportation-type distance $\bar{\mathsf d}{{\boldsymbol\Upsilon}}$; (c) the irreducibility of $\mu$-symmetric Dirichlet forms on ${\boldsymbol\Upsilon}$. 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 $\mathrm{sine}{2}$, $\mathrm{Airy}{2}$, $\mathrm{Bessel}{\alpha, 2}$ ($\alpha \ge 1$), 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.