Equilibrium Kawasaki dynamics and determinantal point processes

TL;DR

This paper develops a general framework for constructing equilibrium Kawasaki dynamics for point processes on discrete spaces, focusing on determinantal processes with specific invariance properties, and introduces a new family of such processes.

Contribution

It provides a novel scheme for equilibrium Kawasaki dynamics under quasi-invariance and identifies a new two-parameter family of determinantal point processes with these properties.

Findings

Constructed a general scheme for equilibrium Kawasaki dynamics.

Identified a two-parameter family of determinantal point processes.

Demonstrated invariance and quasi-invariance properties of these processes.

Abstract

Let "mu" be a point process on a countable discrete space "X". Under assumption that "mu" is quasi-invariant with respect to any finitary permutation of "X", we describe a general scheme for constructing an equilibrium Kawasaki dynamics for which "mu" is a symmetrizing (and hence invariant) measure. We also exhibit a two-parameter family of point processes "mu" possessing the needed quasi-invariance property. Each process of this family is determinantal, and its correlation kernel is the kernel of a projection operator in the Hilbert space of square-summable functions on "X".