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In this paper, we provide relations among the following properties: (a) thetail triviality of a probability measure $\mu$ on the configuration space${\boldsymbol\Upsilon}$; (b) the finiteness of the $L^2$-transportation-typedistance $\bar{\mathsf d}_{{\boldsymbol\Upsilon}}$; (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) ofinteracting infinite diffusions having logarithmic interaction arisen fromdeterminantal/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 DysonBrownian motion is covered. For the proof, the number rigidity of pointprocesses in the sense of Ghosh--Peres plays a key role.

On the Ergodicity of Interacting Particle Systems under Number Rigidity