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The fluctuations of the overlap in the Hopfield model with finitely many patterns at the critical temperature
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  • Published: October 1999

The fluctuations of the overlap in the Hopfield model with finitely many patterns at the critical temperature

  • Barbara Gentz1 &
  • Matthias Löwe3 

Probability Theory and Related Fields volume 115, pages 357–381 (1999)Cite this article

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  • 9 Citations

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Abstract.

We investigate the limiting fluctuations of the order parameter in the Hopfield model of spin glasses and neural networks with finitely many patterns at the critical temperature 1/β c = 1. At the critical temperature, the measure-valued random variables given by the distribution of the appropriately scaled order parameter under the Gibbs measure converge weakly towards a random measure which is non-Gaussian in the sense that it is not given by a Dirac measure concentrated in a Gaussian distribution. This remains true in the case of β = β N →β c = 1 as N→∞ provided β N converges to β c = 1 fast enough, i.e., at speed ?(1/). The limiting distribution is explicitly given by its (random) density.

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Authors and Affiliations

  1. Departement Mathematik, ETH Zürich, ETH Zentrum, CH-8092 Zürich, Switzerland, , , , , , CH

    Barbara Gentz

  2. Fakultät für Mathematik, Universität Bielefeld, Postfach 100131, D-33501 Bielefeld, Germany. e-mail: loewe@mathematik.uni-bielefeld.de, , , , , , DE

    Matthias Löwe

Authors
  1. Barbara Gentz
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  2. Matthias Löwe
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Received: 12 May 1998 / Revised version: 14 October 1998

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Gentz, B., Löwe, M. The fluctuations of the overlap in the Hopfield model with finitely many patterns at the critical temperature. Probab Theory Relat Fields 115, 357–381 (1999). https://doi.org/10.1007/s004400050241

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  • Issue Date: October 1999

  • DOI: https://doi.org/10.1007/s004400050241

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  • Mathematics Subject Classification (1991): 60F05, 60K35 (primary), 82C32 (secondary)
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