Dyson models under renormalization and in weak fields

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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 1/|i-j|α mainly focusing on the range of slow decays 1 < α≤ 2. 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 (|i|+1)⁻ γ. We study how the critical decay power of the field, γ, for which the phase transition persists and the decay power α 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.