A note on equilibrium Glauber and Kawasaki dynamics for permanental point processes

TL;DR

This paper constructs and analyzes equilibrium Glauber and Kawasaki dynamics for permanental point processes, including diffusion approximations leading to interacting Brownian particles, advancing understanding of infinite particle systems in mathematical physics.

Contribution

It introduces new equilibrium dynamics for permanental point processes, including a diffusion approximation for Kawasaki dynamics in Euclidean space.

Findings

Glauber dynamics as a birth-death process with invariance of the permanental point process

Kawasaki dynamics involving particle hopping with invariance properties

Diffusion approximation leading to interacting Brownian particles with permanental invariant measure

Abstract

We construct two types of equilibrium dynamics of an infinite particle system in a locally compact metric space $X$ for which a permanental point process is a symmetrizing, and hence invariant measure. The Glauber dynamics is a birth-and-death process in $X$, while in the Kawasaki dynamics interacting particles randomly hop over $X$. In the case $X=\mathbb R^d$, we consider a diffusion approximation for the Kawasaki dynamics at the level of Dirichlet forms. This leads us to an equilibrium dynamics of interacting Brownian particles for which a permanental point process is a symmetrizing measure.