Abstract
We study the conditional probabilities of the Curie-Weiss Ising model in vanishing external field under a symmetric independent stochastic spin-flip dynamics and discuss their set of points of discontinuity (bad points). We exhibit a complete analysis of the transition between Gibbsian and non-Gibbsian behavior as a function of time, extending the results for the corresponding lattice model, where only partial answers can be obtained. For initial temperature \(\beta^{-1}\,{\geq}\,1\), we prove that the time-evolved measure is always Gibbsian. For \(\frac{2}{3}\,{\leq}\,\beta^{-1}{ < }1\), the time-evolved measure loses its Gibbsian character at a sharp transition time. For \(\beta^{-1}\,{ < }\,\frac{2}{3}\), we observe the new phenomenon of symmetry-breaking in the set of points of discontinuity: Bad points corresponding to non-zero spin-average appear at a sharp transition time and give rise to biased non-Gibbsianness of the time-evolved measure. These bad points become neutral at a later transition time, while the measure stays non-Gibbs. In our proof we give a detailed description of the phase-diagram of a Curie-Weiss random field Ising model with possibly non-symmetric random field distribution based on bifurcation analysis.
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Communicated by J.L. Lebowitz
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Külske, C., Le Ny, A. Spin-Flip Dynamics of the Curie-Weiss Model: Loss of Gibbsianness with Possibly Broken Symmetry. Commun. Math. Phys. 271, 431–454 (2007). https://doi.org/10.1007/s00220-007-0201-y
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DOI: https://doi.org/10.1007/s00220-007-0201-y