Annales de l'Institut Henri Poincaré, Probabilités et Statistiques

On the mixed even-spin Sherrington–Kirkpatrick model with ferromagnetic interaction

Wei-Kuo Chen

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Abstract

We study a spin system with both mixed even-spin Sherrington–Kirkpatrick (SK) couplings and Curie–Weiss (CW) interaction. Our main results are: (i) The thermodynamic limit of the free energy is given by a variational formula involving the free energy of the SK model with a change in the external field. (ii) In the presence of a centered Gaussian external field, the positivity of the overlap and the extended Ghirlanda–Guerra identities hold on a dense subset of the temperature parameters. (iii) We establish a general inequality between the magnetization and overlap. (iv) We construct a temperature region in which the magnetization can be quantitatively controlled and deduce different senses of convergence for the magnetization depending on whether the external field is present or not. Our approach is based on techniques from the study of the CW and SK models and results in convex analysis.

Résumé

Nous étudions un système dont les spins ont à la fois des couplages du type Sherrington–Kirkpatrick (SK) et des interactions du type Curie–Weiss (CW). Nos principaux résultats sont les suivants : (i) la limite thermodynamique de l’énergie libre est donnée par une formule variationnelle impliquant l’énergie libre du modèle SK avec un changement dans le champ magnétique externe. (ii) En présence d’un champ extérieur Gaussien centré, le recouvrement est positif et les identités généralisées de Ghirlanda–Guerra sont valides pour un sous ensemble dense des paramètres de température. (iii) Nous établissons une inégalité générale entre l’aimantation et le recouvrement. (iv) Nous identifions un domaine de températures où l’aimantation peut être contrôlée quantitativement et nous déduisons plusieurs types de convergences pour l’aimantation en présence ou non d’un champ extérieur. Notre approche repose sur des méthodes développées pour l’étude des modèles CW et SK et sur des résultats d’analyse convexe.

Article information

Source
Ann. Inst. H. Poincaré Probab. Statist., Volume 50, Number 1 (2014), 63-83.

Dates
First available in Project Euclid: 1 January 2014

Permanent link to this document
https://projecteuclid.org/euclid.aihp/1388545265

Digital Object Identifier
doi:10.1214/12-AIHP521

Mathematical Reviews number (MathSciNet)
MR3161522

Zentralblatt MATH identifier
1290.60101

Subjects
Primary: 60K35: Interacting random processes; statistical mechanics type models; percolation theory [See also 82B43, 82C43] 82B44: Disordered systems (random Ising models, random Schrödinger operators, etc.)

Keywords
Ferromagnetic interaction Ghirlanda–Guerra identities Parisi formula Sherrington–Kirkpatrick model Ultrametricity

Citation

Chen, Wei-Kuo. On the mixed even-spin Sherrington–Kirkpatrick model with ferromagnetic interaction. Ann. Inst. H. Poincaré Probab. Statist. 50 (2014), no. 1, 63--83. doi:10.1214/12-AIHP521. https://projecteuclid.org/euclid.aihp/1388545265


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