A class of nonergodic interacting particle systems with unique invariant measure

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 ${\mu }_{\phi }^{\prime }$ labeled by the uncountably many values of $\phi$ in the one-dimensional sphere (introduced by van Enter, Opoku, Külske [J. Phys. A 44 (2011) 475002, 11]). In the present paper we construct an associated Markov jump process with quasilocal rates whose semigroup $(S_t{)}_{t\ge 0}$ acts by a continuous rotation $S_t({\mu }_{\phi }^{\prime })={\mu }_{\phi +t}^{\prime }$. 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.

Published in at http://dx.doi.org/10.1214/13-AAP987 the Annals of Applied Probability (http://www.imstat.org/aap/) by the Institute of Mathematical Statistics (http://www.imstat.org)

References in corpus (5)

• Relative entropy and variational properties of generalized Gibbsian measures
• A non-ergodic probabilistic cellular automaton with a unique invariant measure
• Discrete approximations to vector spin models
• Rotating states in driven clock- and XY-models
• Synchronization for discrete mean-field rotators

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• Gibbsian and non-Gibbsian properties of the generalized mean-field fuzzy Potts-model
• Synchronization for discrete mean-field rotators
• Attractor properties of non-reversible dynamics w.r.t invariant Gibbs measures on the lattice
• On the long-time behaviour of reversible interacting particle systems in one and two dimensions
• Non-robust phase transitions in the generalized clock model on trees
• Time-periodic behaviour in one- and two-dimensional interacting particle systems