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.
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