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Correlated Random Systems: Five Different Methods pp 45–70Cite as

Spontaneous Replica Symmetry Breaking and Interpolation Methods for Complex Statistical Mechanics Systems

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Part of the book series: Lecture Notes in Mathematics ((LNM,volume 2143))

Abstract

The phenomenon of spontaneous replica symmetry breaking, for some mean field models for spin glasses, as the celebrated Sherrington-Kirkpatrick model, was discovered by Giorgio Parisi, in the frame of the so called “replica trick”, where the number of replicas goes to zero. Quite recently a rigorous treatment has been possible by using interpolation methods. Interpolation is a very powerful instrument. We give many examples of its use. In these lectures we give a short review about spontaneous replica symmetry breaking for an integer number of replicas, by exploiting the simple Random Energy Model. Moreover, we show how interpolation methods work in the treatment of neural networks models, in particular for the replica symmetric approximation. Finally, we apply interpolation methods to relate various instances of multi-species models, where for example mean field spin glasses are made to interact through a multi-partite interaction. As an application, we get a very simple control on the whole ergodic region for a class of neural networks. Some conclusion and hints for future developments are finally presented.

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Author information

Authors and Affiliations

  1. Dipartimento di Fisica, Università di Roma “La Sapienza”, Roma, Italy

    Francesco Guerra

  2. Istituto Nazionale di Fisica Nucleare, Sezione di Roma, Piazzale A. Moro 5, 00185, Roma, Italy

    Francesco Guerra

Authors
  1. Francesco Guerra
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Correspondence to Francesco Guerra .

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Editors and Affiliations

  1. Institut de Mathématiques de Marseille, Aix-Marseille Université CNRS, Marseille Cedex 13, France

    Véronique Gayrard

  2. Institut für Mathematik, Goethe-Universität Frankfurt, Frankfurt am Main, Germany

    Nicola Kistler

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Guerra, F. (2015). Spontaneous Replica Symmetry Breaking and Interpolation Methods for Complex Statistical Mechanics Systems. In: Gayrard, V., Kistler, N. (eds) Correlated Random Systems: Five Different Methods. Lecture Notes in Mathematics, vol 2143. Springer, Cham. https://doi.org/10.1007/978-3-319-17674-1_2

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