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

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 \(J_{x,y} = J(|x-y|)\equiv \frac{1}{|x-y|^{2-\alpha }}\) with \(\alpha \in [0, 1)\), 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 \(\alpha \ne 0\), proposed by Cassandro, Ferrari, Merola and Presutti. As proved by Fröhlich and Spencer for \(\alpha =0\) and conjectured by Cassandro et al for the region they could treat, \(\alpha \in (0,\alpha _{+})\) for \(\alpha _+=\log (3)/\log (2)-1\), although in the literature dealing with contour methods for these models it is generally assumed that \(J(1)\gg 1\), 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 \(\alpha \in [0,1)\). Moreover, we show that when we add a magnetic field decaying to zero, given by \(h_x= h_*\cdot (1+|x|)^{-\gamma }\) and \(\gamma >\max \{1-\alpha , 1-\alpha ^* \}\) where \(\alpha ^*\approx 0.2714\), the transition still persists.