Comptes Rendus
no. 4
Mathematical Physics/Probability Theory
On the distribution of the overlaps at given disorder
[Sur la distribution des recouvrements à désordre donné.]

Présenté par : Gilles Pisier

Giorgio Parisi1 ; Michel Talagrand2
1 Center for Statistical Mechanics and Complexity, INFM Roma “la Sapienza”, Piazzale Aldo Morro 2, 00185 Roma, Italy
2 Équipe d'Analyse de l'Institut Mathématique, 4, place Jussieu, 75230 Paris cedex 05, France
Comptes Rendus. Mathématique, Volume 339 (2004) no. 4, pp. 303-306.

Nous prouvons une version rigoureuse du fait suivant. Dans un modèle de verres de spins qui satisfait les identités de Ghirlanda–Guerra générales, à désordre donné, la distribution du recouvrement de deux configurations est discrète, et son support est non-aléatoire.

We prove a rigorous version of the following heuristic statement: if, in a spin glass model, the extended Ghirlanda–Guerra identities are valid, at given disorder the distribution of the overlap of two configurations is discrete, and its support (the smallest closed set that carries this distribution) is non-random.

Reçu le :
Accepté le :
Publié le :
DOI : 10.1016/j.crma.2004.06.014
Giorgio Parisi 1 ; Michel Talagrand 2

1 Center for Statistical Mechanics and Complexity, INFM Roma “la Sapienza”, Piazzale Aldo Morro 2, 00185 Roma, Italy
2 Équipe d'Analyse de l'Institut Mathématique, 4, place Jussieu, 75230 Paris cedex 05, France 
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Giorgio Parisi; Michel Talagrand. On the distribution of the overlaps at given disorder. Comptes Rendus. Mathématique, Volume 339 (2004) no. 4, pp. 303-306. doi : 10.1016/j.crma.2004.06.014. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2004.06.014/

[1] S. Ghirlanda; F. Guerra General properties of overlap distributions in disordered spin systems. Towards Parisi ultrametricity, J. Phys. A, Volume 31 (1998) no. 46, pp. 9149-9155

[2] M. Mézard; G. Parisi; M. Virasoro Spin Glass Theory and Beyond, World Scientific, Singapore, 1987

[3] G. Parisi A sequence of approximate solutions to the S–K model for spin glasses, J. Phys. A, Volume 13 (1980), p. 115

[4] G. Parisi Field Theory, Disorder, Simulation, World Sci. Lecture Notes Phys., vol. 45, World Scientific, Singapore, 1992

[5] M. Talagrand Rigorous low temperature results for the mean field p-spin interaction model, Probab. Theory Related Fields, Volume 117 (2000), pp. 303-360

[6] M. Talagrand Spin Glasses, A Challenge to Mathematicians, Springer-Verlag, 2003

[7] M. Talagrand, The free energy of the mean field spherical model, Probab. Theory Related Fields, in press

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