Physical phenomena commonly observed in nature such as phase transitions, critical phenomena and metastability when studied froma mathematical point of view may give arise to a rich variety of behavior whose study becomes interesting in itself. In Chapter 1 we illustrate the phase transition phenomenon at low temperatures for one-dimensional long range Ising models with inhomogeneous external fields. More precisely, we consider Ising spins arranged on the one-dimensional integer lattice where such spins interact via ferromagnetic pairwise interactions whose strength is inversely proportional to their distance to the power ®; furthermore, the system is put under the influence of an external magnetic field that vanishes with polynomial power ± as the distance between the spin and the origin increases. In that case we show that a phase transition manifests itself in the form of the existence of two distinct infinite-volume Gibbs states, obtained by means of the application of the thermodynamic limit considering “plus” and “minus” boundary conditions respectively, whenever the system is subject at low temperatures and an inequality involving ® and ± holds. The proof of this result is done by means of the Peierls’ contour argument adapted to one-dimensional long range Ising models, first introduced by J. Fröhlich and T. Spencer in 1982 and later modified by M. Cassandro, P.A. Ferrari, I. Merola and E. Presutti in 2005. Our results improve the one obtained by the latter authors since we managed to avoid the assumption of large nearest-neighbor interactions and added the influence of an external field, showing an interplay between the constants ® and ± in order to guarantee the manifestation of the phase transition. @en

We consider one-dimensional long-range spin models (usually called Dyson models), consisting of Ising ferromagnets with slowly decaying long-range pair potentials of the form (formula presented), mainly focusing on the range of slow decays (formula presented). We describe two recent results, one about renormalization and one about the effect of external fields at low temperature. The first result states that a decimated long-range Gibbs measure in one dimension becomes non-Gibbsian, in the same vein as comparable results in higher dimensions for short-range models. The second result addresses the behaviour of such models under inhomogeneous fields, in particular external fields which decay to zero polynomially as (formula presented). We study how the critical decay power of the field, (formula presented), for which the phase transition persists and the decay power (formula presented) of the Dyson model compare, extending recent results for short-range models on lattices and on trees. We also briefly point out some analogies between these results.

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In this note we study metastability phenomena for a class of long-range Ising models in one-dimension. We prove that, under suitable general conditions, the configuration - 1 is the only metastable state and we estimate the mean exit time. Moreover, we illustrate the theory with two examples (exponentially and polynomially decaying interaction) and we show that the critical droplet might be macroscopic or mesoscopic, according to the value of the external magnetic field.

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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 h_x= h_∗· (1 +

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