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 long-range one-dimensional Ising ferromagnets, in which the interaction is given by $J_{x,y} = J(|x-y|)\equiv \frac{1}{|x-y|^{2-α}}$ with $α\in [0, 1)$, 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 $α\neq 0$, 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, $α\in (0,α_{+})$ for $α_+=\log(3)/\log(2)-1$, although in the literature dealing with contour methods for these models it is generally assumed that $J(1)\gg1$, we can 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 $α\in [0,1)$. Moreover, we show that when we add a magnetic field decaying to zero, given by $h_x= h_*\cdot(1+|x|)^{-γ}$ and $γ>\max\{1-α, 1-α^* \}$ where $α^*\approx 0.2714$, the transition still persists.