The p spin interaction model with external field

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The p-spin interaction model with external field by Xavier Bardina1, David Marquez-Carreras2, Carles Rovira2 and Samy Tindel3 1 Departament de Matematiques, Universitat Autonoma de Barcelona 08193 Bellaterra, Barcelona, Spain e-mail: 2 Facultat de Matematiques, Universitat de Barcelona, Gran Via 585, 08007-Barcelona, Spain e-mail: , 3 Departement de Mathematiques, Institut Galilee - Universite Paris 13, Avenue J. B. Clement, 93430-Villetaneuse, France e-mail: Abstract This paper is devoted to a detailed study of a p-spins interaction model with external field, including some sharp bounds on the speed of self averaging of the overlap as well as a central limit theorem for its fluctuations, the thermodynamical limit for the free energy and the definition of an Almeida-Thouless type line. Those results show that the external field dominates the tendency to disorder induced by the increasing level of interaction between spins, and our system will share many of its features with the SK model, which is certainly not the case when the external magnetic field vanishes. 1Partially supported by DGES grants BFM2000-0009, BFM2000-0607, HF2000-0002. 2Partially supported by DGES grants BFM2000-0607 and HF2000-0002 .

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spin configuration

spin

central limit theorem

standard gaussian

interactions requires

self averaging

random variable

magnetic field

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The p-spin interaction model with external eld

Xavier Bardina1,quezMarrera-CarsDdiva2, Carles Rovira2 and Samy Tindel3 1,snUviremtaqieuonomadesitatAutacraBnoleDematrapeetaMedtn 08193 Bellaterra, Barcelona, Spain e-mail: bardina@mat.uab.es 2,tasieBtdcearnaloetaMedtatlucaFerivUns,ueiqatm Gran Via 585, 08007-Barcelona, Spain e-mail: marquez@mat.ub.es, rovira@mat.ub.es 33,s1eitriPainU-srevilaGeelDeu,staqititunItsentdrtemhemeMat epa AvenueJ.B.Clement,93430-Villetaneuse,France e-mail: tindel@math.univ-paris13.fr

Abstract This paper is devoted to a detailed study of ap-spins interaction model with external eld, including some sharp bounds on the speed of self averaging of the overlap as well as a central limit theorem for its uctuations, the thermodynamical limit for the free energy and the denition of an Almeida-Thouless type line. Those results show that the external eld dominates the tendency to disorder induced by the increasing level of interaction between spins, and our system will share many of its features with the SK model, which is certainly not the case when the external magnetic eld vanishes.

1Partially supported by DGES grants BFM2000-0009, BFM2000-0607, HF2000-0002. 2Partially supported by DGES grants BFM2000-0607 and HF2000-0002 .