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Large deviation principles for the Hopfield model and the Kac-Hopfield model
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  • Published: December 1995

Large deviation principles for the Hopfield model and the Kac-Hopfield model

  • Anton Bovier1,
  • Véronique Gayrard2 &
  • Pierre Picco2 

Probability Theory and Related Fields volume 101, pages 511–546 (1995)Cite this article

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Summary

We study the Kac version of the Hopfield model and prove a Lebowitz-Penrose theorem for the distribution of the overlap parameters. At the same time, we prove a large deviation principle for the standard Hopfield model with infinitely many patterns.

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References

  1. Amit, D.J., Gutfreund, H., Sompolinsky, H.: Statistical mechanics of neural networks near saturation. Ann. Phys.173 30–67 (1987)

    Google Scholar 

  2. Bovier, A., Gayrard, V.: Rigorous results on the thermodynamics of the dilute Hopfield model. J. Stat. Phys.69 597–627 (1993)

    Google Scholar 

  3. Bovier, A., Gayrard, V.: Rigorous results on the Hopfield model of neural networks. Resenhas do IME-USP1 161–172 (1994)

    Google Scholar 

  4. Bovier, A., Gayrard, V., Picco, P.: Gibbs states of the Hopfield model in the regime of perfect memory. Probab. Theory Relat. Fields100 329–363 (1994)

    Google Scholar 

  5. Bovier, A., Gayrard, V., Picco, P.: (in preparation)

  6. Comets, F.: Large deviation estimates for a conditional probability distribution. Applications to random Gibbs measures. Probab. Theory Relat. Fields80 407–432 (1989)

    Google Scholar 

  7. Cassandro, M., Orlandi, E., Presutti, E.: Interfaces and typical Gibbs configurations for one-dimensional Kac potentials. Probab. Theory Relat. Fields96 57–96 (1993)

    Google Scholar 

  8. Deuschel, J.-D., Stroock, D.: Large deviations. Boston: Academic Press 1989

    Google Scholar 

  9. Dawson, D.A., Gärtner, J.: Large deviations from the McKeane-Vlasov limit for weakly interacting diffusions. Stochastics20 247–308 (1987)

    Google Scholar 

  10. Ellis, R.S.: Entropy, large deviations, and statistical mechanics. Berlin Heidelberg New York: 1985

  11. Pastur, L.A., Figotin, A.L.: Exactly soluble model of a spin glass. Sov. J. Low Temp. Phys.3(6) 378–383 (1977)

    Google Scholar 

  12. Pastur, L.A., Figotin, A.L.: On the theory of disordered spin systems. Theor. Math. Phys.35 403–414 (1978)

    Google Scholar 

  13. Pastur, L.A., Figotin, A.L.: Infinite range limit for a class of disordered spin systems. Theor. Math. Phys.51 564–569 (1982)

    Google Scholar 

  14. Gayrard, V.: The thermodynamic limit of theq-state Potts-Hopfield model with infinitely many patterns. J. Stat. Phys.68: 977–1011 (1992)

    Google Scholar 

  15. Gärtner, J.: Large deviations from the invariant measure. Theory Probab. Appl.22 24–39 (1977)

    Google Scholar 

  16. Grensing, D., Kühn, K.: On classical spin-glass models. J. Phys.48, 713–721 (1987)

    Google Scholar 

  17. van Hemmen, J.L.: Spin glass model of a neural network. Phys. Rev. A34, 3435–3445 (1986)

    Google Scholar 

  18. Hopfield, J.J.: Neural networks and physical systems with emergent collective computational abilities. Proc. Natl. Acad. Sci. USA79, 2554–2558 (1982)

    Google Scholar 

  19. Koch, H.: A free energy bound for the Hopfield model. J. Phys. A: Math Gen.26, L353-L355 (1993)

    Google Scholar 

  20. Koch, H., Piasko, J.: Some rigorous results on the Hopfield neural network model. J. Stat. Phys.55, 903–928 (1989)

    Google Scholar 

  21. Kesten, H., Schonmann, R.: Behaviour in large dimensions of the Potts and Heisenberg model, Rev. Math. Phys.1, 147–182 (1990)

    Google Scholar 

  22. Kac, M., Uhlenbeck, G. and Hemmer, P.C.: On the van der Waals theory of vapour-liquid equilibrium. I. Discussion of a one-dimensional model. J. Math. Phys.4, 216–228 (1963); II. Discussion of the distribution functions, J. Math. Phys.4, 229–247 (1963); III. Discussion of the critical region, J. Math. Phys.5, 60–74 (1964)

    Google Scholar 

  23. Lebowitz, J., Penrose, O.: Rigorous treatment of the Van der Waals Maxwell theory of the liquid-vapour transition. J. Math. Phys.7, 98–113 (1966)

    Google Scholar 

  24. Ledoux, M., Talagrand, M.: Probability in Banach spaces, Springer, Berlin-Heidelberg-New York, 1991

    Google Scholar 

  25. Pastur, L., Shcherbina, M.: Absence of self-averaging of the order parameter in the Sherrington-Kirkpatrick model, J. Stat. Phys.62, 1–19 (1991)

    Google Scholar 

  26. Pastur, L., Shcherbina, M. and Tirozzi, B.: The replica symmetric solution without the replica trick for the Hopfield model. J. Stat. Phys.74, 1161–1183 (1994)

    Google Scholar 

  27. Shcherbina, M., Tirozzi, B.: The free energy for a class of Hopfield models. J. Stat. Phys.72, 113–125 (1992)

    Google Scholar 

  28. Yurinskii, V.V.: Exponential inequalities for sums of random vectors. J. Multivariate. Anal.6, 473–499 (1976)

    Google Scholar 

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Author information

Authors and Affiliations

  1. Weierstraß-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. Large deviation principles for the Hopfield model and the Kac-Hopfield model. Probab. Th. Rel. Fields 101, 511–546 (1995). https://doi.org/10.1007/BF01202783

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  • Received: 27 April 1994

  • Issue Date: December 1995

  • DOI: https://doi.org/10.1007/BF01202783

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Mathematics Subject Classification

  • 60K35
  • 82B44
  • 82C32
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