Metastates in disordered mean-field models: Random field and Hopfield models
type: Item
instance of: scholarly article
title: Metastates in disordered mean-field models: Random field and Hopfield models
zbMATH Open document ID: 0924.60092
DOI: 10.1007/BF02732434
author: Christof Külske
published in: Journal of Statistical Physics
publication date: 1999-11-07T00:00:00Z
review text: Statistical mechanics of disordered systems deals with the construction of Gibbs probability measures that are themselves random variables on some probability space describing the `quenched disorder', i.e. typically the random interactions of the system. On of the main issues is to describe the thermodynamic limit, i.e. the limiting measures obtained by taking the system size to infinity. In the random case, what is the most appropriate probabilistic setting to formulate this problem? This important and long neglected issue has been addressed in a series of papers by Ch. Newman and D. Stein [see, e.g., \textit{Ch. M. Newman}, ``Topics in disordered systems'' (1997; Zbl 0897.60093)] who proposed two concepts of `metastates' (probability states on the space of Gibbs measures) as appropriate candidates to study the infinite volume limits of disordered systems. In the present paper the author gives explicit constructions of the two types of metastates proposed in the simplest examples: the random field version of the Curie-Weiss model, and the Hopfield model with finitely many patterns. It turns out that already in this small class of models, a variety of behaviours can be found, and a lot of insight in the general concept can be gained (the author even exhibits a counterexample to a conjecture of Newman and Stein concerning the relation of their two types of metastates). The reviewer would recommend reading of this paper to anyone interested in understanding the metastate concept.
reviewed by: Anton Bovier
Mathematics Subject Classification ID: 60K40
Mathematics Subject Classification ID: 82B44
Mathematics Subject Classification ID: 60G57
zbMATH DE Number: 1279580
zbMATH Keywords: disordered system
zbMATH Keywords: random Gibbs measures
zbMATH Keywords: metastates
zbMATH Keywords: mean field models
zbMATH Keywords: Hopfield model
zbMATH Keywords: random field model
MaRDI profile type: Publication
cites work: Random infinite-volume Gibbs states for the Curie-Weiss random field Ising model
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cites work: Rounding effects of quenched randomness on first-order phase transitions
cites work: The retrieval phase of the Hopfield model: A rigorous analysis of the overlap distribution
cites work: Q4379449
cites work: Gibbs states of the Hopfield model in the regime of perfect memory
cites work: Large deviation estimates for a conditional probability distribution. Applications to random interaction Gibbs measures
cites work: Q4692653
cites work: Q3134548
cites work: Q5512461
cites work: An almost sure central limit theorem for the overlap parameters in the Hopfield model
cites work: Gibbs measures and phase transitions
cites work: Neural networks and physical systems with emergent collective computational abilities.
cites work: Pressure and variational principle for random Ising model
cites work: Q3840949
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cites work: Strong approximation for set-indexed partial-sum processes, via KMT constructions. II
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cites work: Q3997990
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description: scientific article; zbMATH DE number 1279580