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

  • Francesco GuerraEmail author
Part of the Lecture Notes in Mathematics book series (LNM, volume 2143)


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.


Interpolation methods Mean field spin glasses Multi-species models Neural networks Sherrington-Kirkpatrick model Spontaneous replica symmetry breaking 


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© Springer International Publishing Switzerland 2015

Authors and Affiliations

  1. 1.Dipartimento di FisicaUniversità di Roma “La Sapienza”RomaItaly
  2. 2.Istituto Nazionale di Fisica NucleareSezione di RomaRomaItaly

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