Long-range Ising models: Contours, phase transitions and decaying fields

Autor(es)

Affonso, Lucas ; Bissacot, Rodrigo ; Endo, Eric O. ; Handa, Satoshi

• Tipo de documento: Artigo Científico
• Fonte: JOURNAL OF THE EUROPEAN MATHEMATICAL SOCIETY; v. 27, n. 4, p. 36-pg., 2025-01-01.

Resumo

Inspired by Frohlich-Spencer and subsequent authors who introduced the notion of contour for long-range systems, we provide a definition of contour and a direct proof for the phase transition for ferromagnetic long-range Ising models on Z(d), d >= 2. The argument, which is based on a multiscale analysis, works for the sharp region alpha > d and improves previous results obtained by Park for alpha > 3d + 1, and by Ginibre, Grossmann and Ruelle for alpha > d + 1, where alpha is the power of the coupling constant. The key idea is to avoid a large number of small contours. As an application, we prove the persistence of the phase transition when we add a polynomially decaying magnetic field with power delta > 0 as h(& lowast; )divided by x divided by(-delta), where h(& lowast; )> 0. For d < alpha < d + 1, the phase transition occurs when delta > alpha - d, and when h(& lowast;) is small enough over the critical line delta = alpha - d. For alpha >= d + 1, delta > 1 is enough to prove the phase transition, and for delta = 1 we have to ask h(& lowast;) small. The natural conjecture is that this region is also sharp for the phase transition problem when we have a decaying field. (AU)

• Processo FAPESP: 16/25053-8 - Dinâmica e geometria em baixas dimensões
• Beneficiário: André Salles de Carvalho
• Modalidade de apoio: Auxílio à Pesquisa - Temático
• Processo FAPESP: 20/14563-0 - Condição de fronteira e estados KMS
• Beneficiário: Lucas Affonso Silva Pereira
• Modalidade de apoio: Bolsas no Exterior - Estágio de Pesquisa - Doutorado Direto
• Processo FAPESP: 17/18152-2 - Transições de fase em Modelos de Ising quânticos
• Beneficiário: Lucas Affonso Silva Pereira
• Modalidade de apoio: Bolsas no Brasil - Doutorado Direto