On the meaning of Parisi's functional order parameter

Michel Talagrand

doi:10.1016/j.crma.2003.09.013

Mathematical Physics/Probability Theory

On the meaning of Parisi's functional order parameter

Presented by: Yves Meyer

Michel Talagrand ¹

¹ Équipe d'analyse de l'institut mathématique, 4, place Jussieu, 75230 Paris cedex 05, France

Comptes Rendus. Mathématique, Volume 337 (2003) no. 9, pp. 625-628.

Abstract
Résumé

The author has recently proved that a famous formula discovered by G. Parisi gives at any temperature the correct value for the limiting free energy of a large class of mean field models for spin glasses (a class which contains in particular the Sherrington–Kirkpatrick model). Here we prove rigorously that (generically) the “functional order parameter” occuring in this formula can be interpreted as predicted by Parisi, namely as representing the limiting distribution of the overlap of two independent configurations.

L'auteur a récemment démontré qu'une célèbre formule de G. Parisi donne effectivement à toute température la valeur correcte de l'énergie libre limite d'une large classe de modèles de verre de spin à champ moyen, classe contenant en particulier le modèle de Sherrington–Kirkpatrick. Cette formule fait intervenir un « paramètre d'ordre fonctionnel » dont on démontre ici que (génériquement) la signification est celle prévue par la théorie de Parisi, à savoir qu'il représente la distribution limite du recouvrement de deux configurations indépendentes.

Received: 2003-09-14
Accepted: 2003-09-15
Published online: 2003-10-20

DOI: 10.1016/j.crma.2003.09.013

Author's affiliations:

Michel Talagrand ¹

¹ Équipe d'analyse de l'institut mathématique, 4, place Jussieu, 75230 Paris cedex 05, France

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     author = {Michel Talagrand},
     title = {On the meaning of {Parisi's} functional order parameter},
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     pages = {625--628},
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     number = {9},
     year = {2003},
     doi = {10.1016/j.crma.2003.09.013},
     language = {en},
}

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Michel Talagrand. On the meaning of Parisi's functional order parameter. Comptes Rendus. Mathématique, Volume 337 (2003) no. 9, pp. 625-628. doi : 10.1016/j.crma.2003.09.013. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2003.09.013/

References
Cited by

[1] F. Guerra Replica broken bounds in the mean field spin glass model, Comm. Math. Phys., Volume 233 (2003), pp. 1-12

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

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

[4] M. Talagrand The generalized Parisi formula, C. R. Acad. Sci. Paris, Ser. I, Volume 337 (2003), pp. 111-114

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