Comptes Rendus
Probability Theory/Mathematical Physics
The Ghirlanda–Guerra identities for mixed p-spin model
[Les identités de Ghirlanda–Guerra pour les mélanges de modèles à p-spin]
Comptes Rendus. Mathématique, Volume 348 (2010) no. 3-4, pp. 189-192.

Nous montrons que sous les conditions connues pour impliquer la validité de la formule de Parisi, si l'Hamiltonien du modè le générique de Sherrington–Kirkpatrick Hamiltonien contient un « Hamiltonien de p-spin » alors les identités de Ghirlanda–Guerra pour la puissance p des recouvrements sont valides dans un sens fort (et pas seulement en moyenne sur les parametres).

We show that, under the conditions known to imply the validity of the Parisi formula, if the generic Sherrington–Kirkpatrick Hamiltonian contains a p-spin term then the Ghirlanda–Guerra identities for the pth power of the overlap hold in a strong sense without averaging. This implies strong version of the extended Ghirlanda–Guerra identities for mixed p-spin models than contain terms for all even p2 and p=1.

Reçu le :
Accepté le :
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DOI : 10.1016/j.crma.2010.02.004
Dmitry Panchenko 1

1 Department of Mathematics, Texas A&M University, 77843 College Station, TX, USA
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Dmitry Panchenko. The Ghirlanda–Guerra identities for mixed p-spin model. Comptes Rendus. Mathématique, Volume 348 (2010) no. 3-4, pp. 189-192. doi : 10.1016/j.crma.2010.02.004. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2010.02.004/

[1] S. Chatterjee The Ghirlanda–Guerra identities without averaging, 2009 (preprint) | arXiv

[2] S. Ghirlanda; F. Guerra General properties of overlap probability distributions in disordered spin systems. Towards Parisi ultrametricity, J. Phys. A, Volume 31 (1998) no. 46, pp. 9149-9155

[3] D. Panchenko On differentiability of the Parisi formula, Electron. Comm. Probab., Volume 13 (2008), pp. 241-247

[4] G. Parisi A sequence of approximate solutions to the S-K model for spin glasses, J. Phys. A, Volume 13 (1980), p. L-115

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

[6] M. Talagrand Parisi formula, Ann. of Math. (2), Volume 163 (2006) no. 1, pp. 221-263

[7] M. Talagrand, Construction of pure states in mean-field models for spin glasses, preprint (2008), Probab. Theory Related Fields, in press, http://www.springerlink.com/content/y507332m08275t67/?p=d35ca639b02943ecae07559b26ef2abf&pi=10

[8] M. Talagrand, Mean field models for spin glasses, manuscript

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