Ergodicity of unlabeled dynamics of Dyson's model in infinite dimensions

TL;DR

This paper proves that the unlabeled diffusion process of Dyson's infinite-dimensional model with logarithmic interactions is ergodic, confirming long-term statistical stability of the system.

Contribution

It establishes the ergodicity of the unlabeled dynamics of Dyson's model in infinite dimensions, resolving an open problem from previous research.

Findings

Proved ergodicity of Dyson's model in infinite dimensions

Confirmed the long-term stability of the system's statistical properties

Utilized Dirichlet form and sine2 point process as reference measure

Abstract

Dyson's model in infinite dimensions is a system of Brownian particles that interact via a logarithmic potential with an inverse temperature of $ \beta = 2$. The stochastic process can be represented by the solution to an infinite-dimensional stochastic differential equation. The associated unlabeled dynamics (diffusion process) are given by the Dirichlet form with the sine$ _2$ point process as a reference measure. In a previous study, we proved that Dyson's model in infinite dimensions is irreducible, but left the ergodicity of the unlabeled dynamics as an open problem. In this paper, we prove that the unlabeled dynamics of Dyson's model in infinite dimensions are ergodic.