WIAS Preprint No. 353, (1997)

Statistical mechanics of neural networks: The Hopfield model and the Kac-Hopfield model


  • Bovier, Anton
  • Gayrard, Véronique

2010 Mathematics Subject Classification

  • 82B44 82C32 60K35


  • Hopfield model, mean field theory, Kac-models, neural networks, Gibbs measures, large deviations, replica symmetry




We survey the statistical mechanics approach to the analysis of neural networks of the Hopfield type. We consider both models on complete graphs (mean-field), random graphs (dilute model), and on regular lattices (Kac-model). We try to explain the main ideas and techniques, as well as the results obtained by them, without however going into too much technical detail. We also give a short history of the main developments in the mathematical analysis of these models over the last 20 years.

Appeared in

  • Markov Proc. Related Fields, 3 (1997), pp. 393-423

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