WIAS Preprint No. 283, (1996)

An almost sure central limit theorem for the Hopfield model


  • Bovier, Anton
  • Gayrard, Véronique

2010 Mathematics Subject Classification

  • 60F05 60K35


  • Hopfield model, neural networks, central limit theorem, Brascamp-Lieb inequalities




We prove a central limit theorem for the finite dimensional marginals of the Gibbs distribution of the macroscopic 'overlap'-parameters in the Hopfield model in the case where the number of random 'patterns', M, as a function of the system size N satisfies limN↑∞M(N)/N = 0, without any assumptions on the speed of convergence. The covariance matrix of the limiting gaussian distributions is diagonal and independent of the disorder for almost all realizations of the patterns.

Appeared in

  • Markov Proc. Related Fields, 3 (1997), pp. 151-173

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