A special seminar on/at irregular day, time and place.

Venue: Room 111, RIMS

Abstract:
We construct a class of interacting particle systems with the somewhat surprising property of having a unique translation-invariant and invariant measure, which is not long-time limit of all starting measures, related to an old question stated for example in the book of Liggett.

Examples of this phenomenon have been recently given by Maes and Shlosman for a continuous-spin statistical mechanics model driven by Brownian Motion (Langevin dynamics) as well as by Chassaing and Mairesse for a system with synchronous updating with local translation-invariant interactions (probabilistic cellular automata.) The novelty in our example is to present a system with discrete and finite single spin space and translation-invariant quasilocal rates.

All examples rely on the possibility of creating time periodic behavior. We build on the three dimensional lattice rotator model from statistical mechanics in the phase transition region. Under fine enough but finite discretization of the local state space the discretized model inherits the continuum of translation-invariant, extremal Gibbs measures $\mu_\phi$, parametrized by $\phi\in S^1$. Using the Gibbsian specification for the discrete model we manage to define a dynamics with quasilocal translation-invariant rates that rotates the extremal Gibbs states periodically in time. Since $\frac{1}{2\pi}\int_{S^1}\mu_\phi d\phi$ is invariant, we get non-ergodicity. In order to prove uniqueness of the invariant measure we have to complement the rotation dynamics with a Glauber part and use relative entropy methods.