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December 2014 A class of nonergodic interacting particle systems with unique invariant measure
Benedikt Jahnel, Christof Külske
Ann. Appl. Probab. 24(6): 2595-2643 (December 2014). DOI: 10.1214/13-AAP987


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, Külske [J. Phys. A 44 (2011) 475002, 11]). In the present paper we construct an associated Markov jump process with quasilocal rates whose semigroup $(S_{t})_{t\geq0}$ 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.


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Benedikt Jahnel. Christof Külske. "A class of nonergodic interacting particle systems with unique invariant measure." Ann. Appl. Probab. 24 (6) 2595 - 2643, December 2014.


Published: December 2014
First available in Project Euclid: 26 August 2014

zbMATH: 1304.82014
MathSciNet: MR3262512
Digital Object Identifier: 10.1214/13-AAP987

Primary: 60K35, 82B20, 82C22

Rights: Copyright © 2014 Institute of Mathematical Statistics


Vol.24 • No. 6 • December 2014
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