The Annals of Applied Probability

A class of nonergodic interacting particle systems with unique invariant measure

Benedikt Jahnel and Christof Külske

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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.

Article information

Ann. Appl. Probab., Volume 24, Number 6 (2014), 2595-2643.

First available in Project Euclid: 26 August 2014

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Primary: 82B20: Lattice systems (Ising, dimer, Potts, etc.) and systems on graphs 82C22: Interacting particle systems [See also 60K35] 60K35: Interacting random processes; statistical mechanics type models; percolation theory [See also 82B43, 82C43]

Interacting particle systems nonequilibrium nonergodicity discretization Gibbs measures $XY$-model clock model


Jahnel, Benedikt; Külske, Christof. A class of nonergodic interacting particle systems with unique invariant measure. Ann. Appl. Probab. 24 (2014), no. 6, 2595--2643. doi:10.1214/13-AAP987.

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