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Gibbs states of the Hopfield model in the regime of perfect memory
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  • Published: September 1994

Gibbs states of the Hopfield model in the regime of perfect memory

  • Anton Bovier1,
  • Véronique Gayrard2 &
  • Pierre Picco2 

Probability Theory and Related Fields volume 100, pages 329–363 (1994)Cite this article

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  • 34 Citations

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Summary

We study the thermodynamic properties of the Hopfield model of an autoassociative memory. IfN denotes the number of neurons andM (N) the number of stored patterns, we prove the following results: IfM/N↓ 0 asN↑ ∞, then there exists an infinite number of infinite volume Gibbs measures for all temperaturesT<1 concentrated on spin configurations that have overlap with exactly one specific pattern. Moreover, the measures induced on the overlap parameters are Dirac measures concentrated on a single point and the Gibbs measures on spin configurations are products of Bernoulli measures. IfM/N → α, asN↓∞ for α small enough, we show that for temperaturesT smaller than someT(α)<1, the induced measures can have support only on a disjoint union of balls around the previous points, but we cannot construct the infinite volume measures through convergent sequences of measures.

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Author information

Authors and Affiliations

  1. Institut für Angewandte Analysis und Stochastik, Mohrenstrasse 39, D-10117, Berlin, Germany

    Anton Bovier

  2. Centre de Physique Théorique-CNRS, Luminy, Case 907, F-13288, Marseille Cedex 9, France

    Véronique Gayrard & Pierre Picco

Authors
  1. Anton Bovier
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  2. Véronique Gayrard
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  3. Pierre Picco
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Additional information

Work partially supported by the Commission of the European Communities under contract No. SC1-CT91-0695

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Bovier, A., Gayrard, V. & Picco, P. Gibbs states of the Hopfield model in the regime of perfect memory. Probab. Th. Rel. Fields 100, 329–363 (1994). https://doi.org/10.1007/BF01193704

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  • Received: 10 May 1993

  • Revised: 21 April 1994

  • Issue Date: September 1994

  • DOI: https://doi.org/10.1007/BF01193704

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Mathematics Subject Classification (1991)

  • 60K35
  • 82B44
  • 82C32
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