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.