On the ergodicity of interacting particle systems under number rigidity

Abstract

In this paper, we provide relations among the following properties:

1. (a)

   the tail triviality of a probability measure \(\mu \) on the configuration space \({\varvec{\Upsilon }}\);

2. (b)

   the finiteness of a suitable \(L^2\)-transportation-type distance \(\bar{\textsf {d} }_{\varvec{\Upsilon }}\);

3. (c)

   the irreducibility of local \({\mu }\)-symmetric Dirichlet forms on \({\varvec{\Upsilon }}\).

As an application, we obtain the ergodicity (i.e., the convergence to the equilibrium) of interacting infinite diffusions having logarithmic interaction and arising from determinantal/permanental point processes including \(\text {sine}_{2}\), \(\text {Airy}_{2}\), \(\text {Bessel}_{\alpha , 2}\) (\(\alpha \ge 1\)), and \(\text {Ginibre}\) point processes. In particular, the case of the 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.