We consider Dyson models, Ising models with slow polynomial decay, at low temperature and show that its Gibbs measures deep in the phase transition region are not g-measures. The main ingredient in the proof is the occurrence of an entropic repulsion effect, which follows from the mesoscopic stability of a (single-point) interface for these long-range models in the phase transition region.
We consider ferromagnetic long-range Ising models which display phase transitions. They are one-dimensional Ising ferromagnets, in which the interaction is given by Jx,y = J(|x − y|) ≡ 1 |x−y| 2−α with α ∈ [0, 1), in particular, J(1) = 1. For this class of models one way in which one can prove the phase transition is via a kind of Peierls contour argument, using the adaptation of the Fröhlich-Spencer contours for α = 0, proposed by Cassandro, Ferrari, Merola and Presutti. As proved by Fröhlich and Spencer for α = 0 and conjectured by Cassandro et al for the region they could treat, α ∈ (0, α+) for α+ = log(3)/ log(2)−1, although in the literature dealing with contour methods for these models it is generally assumed that J(1) ≫ 1, we will show that this condition can be removed in the contour analysis. In addition, combining our theorem with a recent result of Littin and Picco we prove the persistence of the contour proof of the phase transition for any α ∈ [0, 1). Moreover, we show that when we add a magnetic field decaying to zero, given by hx = h * ·(1+|x|) −γ and γ > max{1 − α, 1 − α * } where α * ≈ 0.2714, the transition still persists.
We consider the ferromagnetic Ising model with spatially dependent external fields on a Cayley tree, and we investigate the conditions for the existence of the phase transition for a class of external fields, asymptotically approaching a homogeneous critical external field. Our results extend earlier results by Rozikov and Ganikhodjaev.
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