On the Ergodicity of Interacting Particle Systems under Number Rigidity

TL;DR

This paper explores the ergodic behavior of interacting particle systems with number rigidity, establishing relations among measure properties and demonstrating convergence to equilibrium for various infinite diffusions related to determinantal point processes.

Contribution

It introduces new relations among measure properties and proves ergodicity for a class of infinite diffusions with logarithmic interactions, including Dyson Brownian motion.

Findings

Ergodicity of infinite diffusions with logarithmic interactions established.

Relations among tail triviality, transportation distance, and irreducibility shown.

Number rigidity is key to proving ergodic behavior.

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.