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One-dimensional Random Field Kac’s Model: Localization of the Phases




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E l e c t r o n i c J o u r n a l o f P r o b a b i l i t y Vol. 10 (2005), Paper no. 24, pages 786-864. Journal URL http://www.math.washington.edu/∼ejpecp/ One-dimensional Random Field Kac’s Model: Localization of the Phases ∗ Marzio Cassandro Dipartimento di Fisica, Università di Roma “La Sapienza”, INFM-Sez. di Roma. P.le A. Moro, 00185 Roma Italy. [email protected] Enza Orlandi Dipartimento di Matematica, Università di Roma Tre, L.go S.Murialdo 1, 00156 Roma, Italy. [email protected] Pierre Picco CPT, UMR CNRS 62072 Luminy, Case 907, F-13288 Marseille Cedex 9, France and CMM, UMR CNRS3 Blanco Encalada 2120, Santiago, Chile. [email protected] Maria Eulalia Vares∗∗ CBPF, Rua Dr. Xavier Sigaud, 150. 22290-180, Rio de Janeiro, RJ,Brasil. [email protected] Abstract We study the typical profiles of a one dimensional random field Kac model, for values of the temperature and magnitude of the field in the region of two absolute minima for the free energy of the corresponding random field Curie Weiss model. We show that, for a set of realizations of the random field of overwhelming probability, the localization of the two phases corresponding to the previous minima is completely determined. Namely, we are able to construct random intervals tagged with a sign, where typically, with respect to the infinite volume Gibbs measure, the profile is rigid and takes, according to the sign, one of the two values corresponding to the previous minima. Moreover, we characterize the transition from one phase to the other. The analysis extends the one done by Cassandro, Orlandi and Picco in [13]. Key Words and Phrases: phase transition, random walk, random environment, Kac potential. AMS 2000 Mathematics Subject Classification: 60K35,82B20,82B43 Submitted to EJP on February 2, 2004. Final version accepted on March 23, 2005. ∗ Supported by: CNR-CNRS-Project 8.005, INFM-Roma; MURST/Cofin 01-02/03-04; FAPERJ Projects E-26/150.940-99 and E-26/151.905/00; CNPq-CNR Project: 91.0069/00-0 2 Université de Provence Aix–Marseille 1, Université de la Mediterranée Aix–Marseille 2 et Université de Toulon et du Var 3 Universidad de Chile ∗∗ Partially supported by CNPq. 14/july/2005; 12:06 786 1 Introduction We consider a one-dimensional spin system interacting via a ferromagnetic two-body Kac potential and external random field given by independent Bernoulli variables. Problems where a stochastic contribution is added to the energy of the system arise naturally in condensed matter physics where the presence of the impurities causes the microscopic structure to vary from point to point. Some of the vast literature on these topics may be seen consulting [1-6], [10], [18-21], [23], [32]. Kac’s potentials is a short way to denote two-body ferromagnetic interactions with range 1γ , where γ is a dimensionless parameter such that when γ → 0, i.e. very long range, the strength of the interaction becomes very weak, but in such a way that the total interaction between one spin and all the others is finite. They were introduced in [22], and then generalized in [24], to provide a rigorous proof of the validity of the van der Waals theory ...





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