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 $J_{x,y}=J(|x-y|)\equiv \frac{1}{|x-y{|}^{2-\alpha }}$ 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 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 $\alpha \in [0,1)$. Moreover, we show that when we add a magnetic field decaying to zero, given by $h_x=h_{\ast }\cdot (1+|x|{)}^{-\gamma }$ and $\gamma >\max \{1-\alpha ,1-{\alpha }^{\ast }\}$ where ${\alpha }^{\ast }\approx 0.2714$, the transition still persists.

13 pages

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Cited by in corpus (1)

• Entropic repulsion and lack of the $g$-measure property for Dyson models