Comptes Rendus
Probability Theory/Mathematical Physics
Rigorous construction of the pure states for certain spin glass models
Comptes Rendus. Mathématique, Volume 346 (2008) no. 17-18, pp. 1003-1005.

For a large class of Gaussian Hamiltonians, that includes the Sherrington–Kirkpatrick model, the p-spin interaction model for even p, and many others, we prove at each temperature (under a very mild and essentially necessary condition) that generically the “pure states” predicted by physics do exist. The configuration space decomposes in an essentially unique manner in a sequence of subsets on which the overlap takes essentially its maximum value.

Pour une large classe d'Hamiltoniens Gaussiens, comprenant le modèle de Sherrington–Kirkpatrick, le modèle à p-spin pour p pair, et bien d'autres, nous prouvons à toute temperature (sous des conditons très peu restrictives) l'existence des « états purs » prédits par les physiciens. Génériquement, à désordre donné, l'espace des configurations se décompose de façon essentiellement unique en une suite de sous-ensembles sur lesquels le recouvrement de deux configurations prend essentiellement sa valeur maximale.

Received:
Accepted:
Published online:
DOI: 10.1016/j.crma.2008.07.008
Michel Talagrand 1

1 Équipe d'analyse de l'institut mathématique, 4, place Jussieu, 75230 Paris cedex 05, France
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Michel Talagrand. Rigorous construction of the pure states for certain spin glass models. Comptes Rendus. Mathématique, Volume 346 (2008) no. 17-18, pp. 1003-1005. doi : 10.1016/j.crma.2008.07.008. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2008.07.008/

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

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

[3] M. Talagrand Parisi measures, J. Funct. Anal., Volume 231 (2006) no. 2, pp. 269-286

[4] M. Talagrand Free energy of the spherical mean field model, Probab. Theory Related Fields, Volume 134 (2006) no. 3, pp. 339-382

[5] M. Talagrand, Mean Field Models for Spin Glasses, book in preparation

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