Annales de l'Institut Henri Poincaré, Probabilités et Statistiques

Some examples of quenched self-averaging in models with Gaussian disorder

Wei-Kuo Chen and Dmitry Panchenko

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Abstract

In this paper we give an elementary approach to several results of Chatterjee in (Disorder chaos and multiple valleys in spin glasses (2013) arXiv:0907.3381, Comm. Math. Phys. 337 (2015) 93–102), as well as some generalizations. First, we prove quenched disorder chaos for the bond overlap in the Edwards–Anderson type models with Gaussian disorder. The proof extends to systems at different temperatures and covers a number of other models, such as the mixed $p$-spin model, Sherrington–Kirkpatrick model with multi-dimensional spins and diluted $p$-spin model. Next, we adapt the same idea to prove quenched self-averaging of the bond magnetization for one system and use it to show quenched self-averaging of the site overlap for random field models with positively correlated spins. Finally, we show self-averaging for certain modifications of the random field itself.

Résumé

Dans cet article, nous présentons une approche élémentaire de plusieurs résultats de Chatterjee (Disorder chaos and multiple valleys in spin glasses (2013) arXiv:0907.3381, Comm. Math. Phys. 337 (2015) 93–102), et quelques généralisations. D’abord, nous prouvons, dans le cas d’un désordre quenched, un résultat de chaos pour le recouvrement par arêtes dans les modèles de type Edwards–Anderson avec désordre gaussien. La preuve s’étend à des systèmes à différentes températures et couvre d’autres modèles comme le modèle $p$-spins mixte, le modèle de Sherrington–Kirkpatrick avec des spins multi-dimensionnels et le modèle $p$-spin dilué. Ensuite, nous adaptons la même idée pour prouver la propriété d’auto-moyennisation du recouvrement par site et nous l’utilisons pour montrer le même résultat pour des modèles avec un champs aléatoire et des spins positivement corrélés. Enfin, nous montrons la propriété d’auto-moyennisation pour certaines modifications du champs aléatoire.

Article information

Source
Ann. Inst. H. Poincaré Probab. Statist., Volume 53, Number 1 (2017), 243-258.

Dates
Received: 24 July 2015
Revised: 8 September 2015
Accepted: 11 September 2015
First available in Project Euclid: 8 February 2017

Permanent link to this document
https://projecteuclid.org/euclid.aihp/1486544891

Digital Object Identifier
doi:10.1214/15-AIHP715

Mathematical Reviews number (MathSciNet)
MR3606741

Zentralblatt MATH identifier
1361.60087

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
Self-averaging Gaussian disorder Spin glasses

Citation

Chen, Wei-Kuo; Panchenko, Dmitry. Some examples of quenched self-averaging in models with Gaussian disorder. Ann. Inst. H. Poincaré Probab. Statist. 53 (2017), no. 1, 243--258. doi:10.1214/15-AIHP715. https://projecteuclid.org/euclid.aihp/1486544891


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References

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