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.
Categories: math.PR math-ph math.FA math.MP