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Possible loss and recovery of Gibbsianness during the stochastic evolution of Gibbs measures

A. C. D. van Enter·R. Fernández·F. den Hollander·F. Redig
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Abstract

We consider Ising-spin systems starting from an initial Gibbs measure $ν$ and evolving under a spin-flip dynamics towards a reversible Gibbs measure $μ\not=ν$. Both $ν$ and $μ$ are assumed to have a finite-range interaction. We study the Gibbsian character of the measure $νS(t)$ at time $t$ and show the following: (1) For all $ν$ and $μ$, $νS(t)$ is Gibbs for small $t$. (2) If both $ν$ and $μ$ have a high or infinite temperature, then $νS(t)$ is Gibbs for all $t>0$. (3) If $ν$ has a low non-zero temperature and a zero magnetic field and $μ$ has a high or infinite temperature, then $νS(t)$ is Gibbs for small $t$ and non-Gibbs for large $t$. (4) If $ν$ has a low non-zero temperature and a non-zero magnetic field and $μ$ has a high or infinite temperature, then $νS(t)$ is Gibbs for small $t$, non-Gibbs for intermediate $t$, and Gibbs for large $t$. The regime where $μ$ 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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A. C. D. van Enter, R. Fernández, F. den Hollander, F. Redig. 2001-05-02. Possible loss and recovery of Gibbsianness during the stochastic evolution of Gibbs measures. https://doi.org/10.1007/s002200200605

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