01-200 A.C.D. van Enter, R. Fern\'andez, F. den Hollander, F. Redig

Possible loss and recovery of Gibbsianness during the stochastic evolution of Gibbs measures (531K, gzipped postscript) May 30, 01

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Abstract. We consider Ising-spin systems starting from an initial Gibbs measure $\nu$ and evolving under a spin-flip dynamics towards a reversible Gibbs measure $\mu neq \nu$. Both $\mu$ and $\nu$ are assumed to have a finite-range interaction. We study the Gibbsian character of the measure $ \nu S(t)$ at time t and show the following: (1) For all $\nu$ and $\mu$, $\nu S(t)$ is Gibbs for small t. (2)If both $\nu$ and $\mu$ have a high or infinite temperature, then $\nu S(t)$ is Gibbs for all $t \geq 0$. (3) If \nu$ has a low non-zero temperature and a zero magnetic field and $\mu$ has a high or infinite temperature, then $\nu S(t)$ is Gibbs for small t and non-Gibbs for large t. (4) If $\nu$ has a low non-zero temperature and a non-zero magnetic field and $\mu$ has a high or infinite temperature, then $\nu S(t)$ is Gibbs for small t, non-Gibbs for intermediate t, and Gibbs for large t. The regime where $\mu$ has a low or zero temperature and t is not small remains open. This regime presumably allows for many different scenarios.

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