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We consider polynomial long-range Ising models in one dimension, with ferromagnetic pair interactions decaying with power 2- (for 0 12 the metastate is still dispersed, but has its support just on the set of the two extremal Gibbs measures, the plus measure and the minus measure. The former, moderate decays case, appears to be new and is due to the occurrence of almost sure boundedness of the random variable which is the sum of all interaction (free) energies between random and ordered half-lines, when the decay is fast enough, but still slow enough to get a phase transition (>0) ; while the latter, slow decays case, is more reminiscent of and similar to the behaviour of higher-dimensional nearest-neighbour Ising models with diverging boundary (free) energies. We leave the threshold case =12 for further studies.
Endo et al. (Tue,) studied this question.