  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.
