TY - JOUR

T1 - Contour Methods for Long-Range Ising Models

T2 - Weakening Nearest-Neighbor Interactions and Adding Decaying Fields

AU - Bissacot, Rodrigo

AU - Endo, Eric O.

AU - van Enter, Aernout C.D.

AU - Kimura, Bruno

AU - Ruszel, Wioletta M.

PY - 2018

Y1 - 2018

N2 - 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 hx= h∗· (1 +

AB - 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 hx= h∗· (1 +

UR - http://www.scopus.com/inward/record.url?scp=85049776312&partnerID=8YFLogxK

U2 - 10.1007/s00023-018-0693-3

DO - 10.1007/s00023-018-0693-3

M3 - Article

AN - SCOPUS:85049776312

VL - 19

SP - 2557

EP - 2574

JO - Annales Henri Poincaré

JF - Annales Henri Poincaré

SN - 1424-0637

IS - 8

ER -