Probability Theory/Mathematical Physics
On Guerra's broken replica-symmetry bound
Comptes Rendus. Mathématique, Volume 337 (2003) no. 7, pp. 477-480.

Consider a random Hamiltonian H N (σ ) for σ Σ N ={0,1} . We assume that the family (H N (σ )) is jointly Gaussian centered and that for σ 1 ,σ 2 Σ N , N -1 EH N (σ 1 )H N (σ 2 ) =ξ(N−1iNσ1iσ2i) for a certain function ξ on . F. Guerra proved the remarkable fact that the free energy of the system with Hamiltonian H N (σ )+h iN σ i is bounded below by the free energy of the Parisi solution provided that ξ is convex on . We prove that this fact remains (asymptotically) true when the function ξ is only assumed to be convex on + . This covers in particular the case of the p-spin interaction model for any p.

Considérons un hamiltonian aléatoire H N (σ )σ Σ N ={0,1} . Nous supposons la famille (H N (σ )) gaussienne centrée et que pour tous σ 1 ,σ 2 Σ N , on ait N -1 EH N (σ 1 )H N (σ 2 )=ξ(N -1 iN σ i 1 σ i 2 ) pour une certaine fonction ξ sur . F. Guerra a prouvé récemment le fait remarquable que l'énergie libre du système d'hamiltonien H N (σ )+h iN σ i est bornée inferieurement par l'énergie libre de la solution de Parisi lorsque ξ est convexe sur . Nous montrons que ceci reste asymptotiquement vrai si l'on suppose seulement que la fonction ξ est convexe sur + . Ce résultat s'applique en particulier au cas du modèle d'interaction à p-spin pour tout p.

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Published online:
DOI: 10.1016/j.crma.2003.09.001
Talagrand, Michel 1

1 Équipe d'analyse de l'institut mathématique, 4, place Jussieu, 75230 Paris cedex 05, France
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Talagrand, Michel. On Guerra's broken replica-symmetry bound. Comptes Rendus. Mathématique, Volume 337 (2003) no. 7, pp. 477-480. doi : 10.1016/j.crma.2003.09.001. http://www.numdam.org/articles/10.1016/j.crma.2003.09.001/

[1] Aizenman, M.; Sims, R.; Starr, S. (An extended variational principle for the SK spin-glass model) | arXiv

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

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

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

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