A class of nonergodic interacting particle systems with unique invariant measure

TL;DR

This paper constructs a Markov jump process for a class of spin models with uncountably many extremal Gibbs measures, demonstrating a unique invariant measure that is not a long-term limit of all initial states.

Contribution

It introduces a novel interacting particle system with a unique invariant measure, addressing a longstanding question in the theory of particle systems.

Findings

Constructed a Markov process with continuous rotation of Gibbs measures.

Provided examples of systems with unique invariant measure not as a long-time limit.

Connected the model to conjectures about the clock model's intermediate temperature regime.

Abstract

We consider a class of discrete $q$-state spin models defined in terms of a translation-invariant quasilocal specification with discrete clock-rotation invariance which have extremal Gibbs measures $\mu'_{\varphi}$ labeled by the uncountably many values of $\varphi$ in the one-dimensional sphere (introduced by van Enter, Opoku, K\"{u}lske [J. Phys. A 44 (2011) 475002, 11]). In the present paper we construct an associated Markov jump process with quasilocal rates whose semigroup $(S_t)_{t\geq0}$ acts by a continuous rotation $S_t(\mu'_{\varphi})=\mu'_{\varphi +t}$. As a consequence our construction provides examples of interacting particle systems with unique translation-invariant invariant measure, which is not long-time limit of all starting measures, answering an old question (compare Liggett [Interacting Particle Systems (1985) Springer], question four, Chapter one). The construction of this particle system is inspired by recent conjectures of Maes and Shlosman about the intermediate temperature regime of the nearest-neighbor clock model. We define our generator of the interacting particle system as a (noncommuting) sum of the rotation part and a Glauber part. Technically the paper rests on the control of the spread of weak nonlocalities and relative entropy-methods, both in equilibrium and dynamically, based on Dobrushin-uniqueness bounds for conditional measures.