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




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J i on Electr o u a rn l o f P c r ob abil ity 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 Universita di Roma La Sapienza INFM Sez di Roma P le A Moro 00185 Roma Italy cassandro roma1 infn it Enza Orlandi Dipartimento di Matematica Universita di Roma Tre L go S Murialdo 1 00156 Roma Italy orlandi mat uniroma3 it Pierre Picco CPT UMR CNRS 6207 Luminy Case 907 F 13288 Marseille Cedex 9 France and CMM UMR CNRS3 Blanco Encalada 2120 Santiago Chile picco cpt univ mrs fr 2 Maria Eulalia Vares CBPF Rua Dr Xavier Sigaud 150 22290 180 Rio de Janeiro RJ Brasil eulalia cbpf br 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 Universite de Provence Aix Marseille 1 Universite de la Mediterrane e Aix Marseille






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