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 ${H}_{N}\left(\stackrel{\to }{\sigma }\right)$ for $\stackrel{\to }{\sigma }\in {\Sigma }_{N}={\left\{0,1\right\}}^{ℕ}.$ We assume that the family $\left({\mathrm{H}}_{N}\left(\stackrel{\to }{\sigma }\right)\right)$ is jointly Gaussian centered and that for ${\stackrel{\to }{\sigma }}^{1},\phantom{\rule{0.277778em}{0ex}}{\stackrel{\to }{\sigma }}^{2}\phantom{\rule{0.277778em}{0ex}}\in {\Sigma }_{N},$ ${N}^{-1}{\mathrm{EH}}_{N}\left({\stackrel{\to }{\sigma }}^{1}\right){\mathrm{H}}_{N}\left({\stackrel{\to }{\sigma }}^{2}\right)$ =ξ(N−1iNσ1iσ2i) for a certain function ξ on $ℝ$. F. Guerra proved the remarkable fact that the free energy of the system with Hamiltonian ${H}_{N}\left(\stackrel{\to }{\sigma }\right)+\mathrm{h}{\sum }_{\mathrm{i}⩽\mathrm{N}}{\sigma }_{i}$ is bounded below by the free energy of the Parisi solution provided that ξ is convex on $ℝ$. We prove that this fact remains (asymptotically) true when the function ξ is only assumed to be convex on ${ℝ}^{+}$. This covers in particular the case of the p-spin interaction model for any p.

Considérons un hamiltonian aléatoire ${H}_{N}\left(\stackrel{\to }{\sigma }\right)$$\stackrel{\to }{\sigma }\in {\Sigma }_{N}={\left\{0,1\right\}}^{ℕ}.$ Nous supposons la famille $\left({\mathrm{H}}_{N}\left(\stackrel{\to }{\sigma }\right)\right)$ gaussienne centrée et que pour tous ${\stackrel{\to }{\sigma }}^{1},\phantom{\rule{3.30002pt}{0ex}}{\stackrel{\to }{\sigma }}^{2}\in {\Sigma }_{N},$ on ait ${N}^{-1}{\mathrm{EH}}_{N}\left({\stackrel{\to }{\sigma }}^{1}\right){\mathrm{H}}_{N}\left({\stackrel{\to }{\sigma }}^{2}\right)=\xi \left({\mathrm{N}}^{-1}{\sum }_{\mathrm{i}⩽\mathrm{N}}{\sigma }_{i}^{1}{\sigma }_{i}^{2}\right)$ pour une certaine fonction ξ sur $ℝ$. F. Guerra a prouvé récemment le fait remarquable que l'énergie libre du système d'hamiltonien ${H}_{N}\left(\stackrel{\to }{\sigma }\right)+\mathrm{h}{\sum }_{\mathrm{i}⩽\mathrm{N}}{\sigma }_{i}$ est bornée inferieurement par l'énergie libre de la solution de Parisi lorsque ξ est convexe sur $ℝ$. Nous montrons que ceci reste asymptotiquement vrai si l'on suppose seulement que la fonction ξ est convexe sur ${ℝ}^{+}$. Ce résultat s'applique en particulier au cas du modèle d'interaction à p-spin pour tout p.

Accepted:
Published online:
DOI: 10.1016/j.crma.2003.09.001

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. 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. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2003.09.001/

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

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

[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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