A coupling construction for spin systems with infinite range interactions

TL;DR

This paper introduces a new coupling construction method for infinite-range spin systems, providing conditions for their well-definedness and invariance, thus advancing the theoretical understanding of such systems.

Contribution

It offers a novel coupling construction approach for infinite-range spin systems and clarifies the natural conditions needed for their well-definedness and invariance.

Findings

Established a sufficient condition for well-defined infinite-range spin systems

Provided a new coupling construction method

Identified criteria for measure invariance

Abstract

Given a countable set of sites and a collection of flip rates at each site, we give a sufficient condition on the long-range dependancies of the flip rates ensuring the well-definedness of the corresponding spin system. This hypothesis has already been widely used since the interacting particle systems were introduced, but our construction brings a new insight to understand why it is natural. The process is first constructed as a limit of finite spin systems. Then we identify its generator and give a simple criterion for a measure to be invariant with respect to it.