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

Autor: Aernout C. D. van Enter, Eric O. Endo, Bruno Kimura, Rodrigo Bissacot, Wioletta M. Ruszel
Přispěvatelé: Dynamical Systems, Geometry & Mathematical Physics, High-Energy Frontier
Jazyk: angličtina
Rok vydání: 2018
Předmět:
Zdroj: Annales Henri Poincaré, 19(8), 2557-2574. Springer International Publishing AG
Repositório Institucional da USP (Biblioteca Digital da Produção Intelectual)
Universidade de São Paulo (USP)
instacron:USP
ISSN: 1424-0637
Popis: 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 Fr\"ohlich-Spencer contours for $\alpha \neq 0$, proposed by Cassandro, Ferrari, Merola and Presutti. As proved by Fr\"ohlich 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 $J(1)\gg1$, 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 $\alpha \in [0,1)$. 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.
Comment: 13 pages
Databáze: OpenAIRE
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