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The Martingale Method for Mean-Field Disordered Systems at High Temperature

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Mathematical Aspects of Spin Glasses and Neural Networks

Part of the book series: Progress in Probability ((PRPR,volume 41))

Abstract

We review some martingale techniques for the study of meanfield disordered systems at high temperature, including the Sherrington-Kirkpatrick model, the Hopfield model, and more general associative memory models. We describe the log-normal limit of the partition function in term of a Brownian motion, and we give another proof of self-averaging for pressure. In the Sherrington-Kirkpatrick case we show that the law of the product of two independent configurations is close to uniform at very high temperature, but this does not persist in the whole high-temperature region.

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Work partially supported by the Commission of the European Union under contract CHRX-CT93-0411

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Author information

Authors and Affiliations

  1. University of California, Irvine, USA

    Francis Comets

  2. UFR de Mathématiques, Université Paris 7 — Denis Diderot, URA CNRS 1321 Modèles aléatoires, case 7012, 2 place Jussieu, 75251, Paris cedex 05, France

    Francis Comets

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  1. Francis Comets
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Editor information

Editors and Affiliations

  1. Weierstrass — Institut für Angewandte Analysis und Stochastik, D-10117, Berlin, Germany

    Anton Bovier

  2. Centre de Physique Théorique, CNRS-Luminy, 13288, Marseille, France

    Pierre Picco

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© 1998 Birkhäuser Boston

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Comets, F. (1998). The Martingale Method for Mean-Field Disordered Systems at High Temperature. In: Bovier, A., Picco, P. (eds) Mathematical Aspects of Spin Glasses and Neural Networks. Progress in Probability, vol 41. Birkhäuser Boston. https://doi.org/10.1007/978-1-4612-4102-7_2

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  • DOI: https://doi.org/10.1007/978-1-4612-4102-7_2

  • Publisher Name: Birkhäuser Boston

  • Print ISBN: 978-1-4612-8653-0

  • Online ISBN: 978-1-4612-4102-7

  • eBook Packages: Springer Book Archive

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