Contour Methods for Long-Range Ising Models: Weakening Nearest-Neighbor Interactions and Adding Decaying Fields

Author(s)

Bissacot, Rodrigo ^[1] ; Endo, Eric O. ^{[1, 2]} ; van Enter, Aernout C. D. ^[2] ; Kimura, Bruno ^[3] ; Ruszel, Wioletta M. ^[3]

Affiliation

^[1] Univ Sao Paulo, IME, Sao Paulo - Brazil

^[2] Univ Groningen, Johann Bernoulli Inst, Groningen - Netherlands

^[3] Delft Univ Technol, Delft Inst Appl Math, Delft - Netherlands

• Document type: Journal article
• Source: ANNALES HENRI POINCARE; v. 19, n. 8, p. 2557-2574, AUG 2018.
• Web of Science Citations: 1

Abstract

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- a with a. {[} 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 Fr <spacing diaeresis> ohlich- Spencer contours for a = 0, proposed by Cassandro, Ferrari, Merola and Presutti. As proved by Fr <spacing diaeresis> ohlich and Spencer for a = 0 and conjectured by Cassandro et al for the region they could treat, a. (0, a+) for a+ = 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 a. {[} 0, 1). Moreover, we show that when we add a magnetic field decaying to zero, given by hx = h{*} center dot (1+| x|) -. and. > max[1- a, 1- a {*}] where a {*} 0.2714, the transition still persists. (AU)

• FAPESP's process: 16/08518-7 - Gibbs Measures and Phase Transitions
• Grantee: Rodrigo Bissacot Proença
• Support Opportunities: Scholarships abroad - Research
• FAPESP's process: 15/14434-8 - Phase Transitions in Spin Models with General External Fields
• Grantee: Eric Ossami Endo
• Support Opportunities: Scholarships abroad - Research Internship - Doctorate
• FAPESP's process: 16/25053-8 - Dynamics and geometry in low dimensions
• Grantee: André Salles de Carvalho
• Support Opportunities: Research Projects - Thematic Grants
• FAPESP's process: 14/10637-9 - Combinatorial Problems in Ferromagnetic Models
• Grantee: Eric Ossami Endo
• Support Opportunities: Scholarships in Brazil - Doctorate