On the longtime behaviour of reversible interacting particle systems in one and two dimensions
Abstract
By refining Holley's free energy technique, we show that, under quite general assumptions on the dynamics, the attractor of a (possibly nontranslationinvariant) interacting particle system in one or two spatial dimensions is contained in the set of Gibbs measures if the dynamics admits a reversible Gibbs measure. In particular, this implies that there can be no reversible interacting particle system that exhibits timeperiodic behaviour and that every reversible interacting particle system is ergodic if and only if the reversible Gibbs measure is unique. In the special case of nonattractive stochastic Ising models this answers a question due to Liggett.
 Publication:

arXiv eprints
 Pub Date:
 March 2023
 DOI:
 10.48550/arXiv.2303.10640
 arXiv:
 arXiv:2303.10640
 Bibcode:
 2023arXiv230310640J
 Keywords:

 Mathematics  Probability;
 Mathematical Physics;
 Primary 82C22;
 Secondary 60K35