Dyson Models Under Renormalization and in Weak Fields

Autor(es)

Bissacot, Rodrigo ; Endo, Eric O. ; van Enter, Aernout C. D. ; Kimura, Bruno ; Le Ny, Arnaud ; Ruszel, Wioletta M. ; Sidoravicius, V

• Tipo de documento: Artigo Científico
• Fonte: SOJOURNS IN PROBABILITY THEORY AND STATISTICAL PHYSICS - I: SPIN GLASSES AND STATISTICAL MECHANICS, A FESTSCHRIFT FOR CHARLES M. NEWMAN; v. 298, p. 15-pg., 2019-01-01.

Resumo

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/vertical bar i - j vertical bar(alpha), mainly focusing on the range of slow decays 1 < alpha <= 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 1 /(vertical bar i vertical bar +1)(gamma). We study how the critical decay power of the field, gamma, for which the phase transition persists and the decay power alpha 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. (AU)

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