Comptes Rendus
no. 17-18
Probability Theory/Mathematical Physics
Rigorous construction of the pure states for certain spin glass models
Gilles Pisier
Michel Talagrand1
1Équipe d'analyse de l'institut mathématique, 4, place Jussieu, 75230 Paris cedex 05, France
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 
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
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[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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