On the distribution of the overlaps at given disorder

Giorgio Parisi; Michel Talagrand

doi:10.1016/j.crma.2004.06.014

Mathematical Physics/Probability Theory

On the distribution of the overlaps at given disorder

Presented by: Gilles Pisier

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

Comptes Rendus. Mathématique, Volume 339 (2004) no. 4, pp. 303-306.

Abstract
Résumé

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.

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.

Received: 2004-05-19
Accepted: 2004-06-09
Published online: 2004-07-28

DOI: 10.1016/j.crma.2004.06.014

Author's affiliations:

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/

References
Cited by

[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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