A CLASS OF NONERGODIC INTERACTING PARTICLE SYSTEMS WITH UNIQUE INVARIANT MEASURE

• 成果类型: Article
• 署名作者: Jahnel, Benedikt; Kuelske, Christof
• 署名单位: Ruhr University Bochum
• 刊物名称: ANNALS OF APPLIED PROBABILITY
• ISSN/ISSBN: 1050-5164
• DOI: 10.1214/13-AAP987
• 发表日期: 2014
• 页码: 2595-2643
• 关键词: equilibrium states; transitions; symmetry; MODEL

摘要

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(1)(phi) labeled by the uncountably many values of phi in the one-dimensional sphere (introduced by van Enter, Opoku, Kulske [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 >= 0) acts by a continuous rotation St (mu(1)(phi)) = mu(1)(phi+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.