Dyson's model in infinite dimensions is irreducible

TL;DR

This paper proves that Dyson's infinite-dimensional Brownian particle system with logarithmic interactions is irreducible, confirming its fundamental stochastic properties and connection to the sine_2 point process.

Contribution

It establishes the irreducibility of Dyson's model in infinite dimensions, a key property for understanding its long-term behavior and mathematical structure.

Findings

Dyson's model is shown to be irreducible in infinite dimensions.

The model's construction via Dirichlet form and sine_2 process is validated.

The results deepen understanding of infinite-dimensional stochastic particle systems.

Abstract

Dyson's model in infinite dimensions is a system of Brownian particles interacting via a logarithmic potential with an inverse temperature of $ \beta = 2$. The stochastic process is given as a solution to an infinite-dimensional stochastic differential equation. Additionally, a Dirichlet form with the sine$ _2$ point process as a reference measure constructs the stochastic process as a functional of the associated configuration-valued diffusion process. In this paper, we prove that Dyson's model in infinite dimensions is irreducible.