WIAS Preprint No. 461, (1998)

Fluctuations in the Hopfield model at the critical temperature


  • Gentz, Barbara
  • Löwe, Matthias

2010 Mathematics Subject Classification

  • 60K35 60F05 82C32


  • Hopfield model, spin glasses, neural networks, random disorder, limit theorems, non-Gaussian fluctuations, critical temperature




We investigate the fluctuations of the order parameter in the Hopfield model of spin glasses and neural networks at the critical temperature 1 / βc = 1. The number of patterns M(N) is allowed to grow with the number N of spins but the growth rate is subject to the constraint M(N)15/N → 0. As the system size N increases, on a set of large probability the distribution of the appropriately scaled order parameter under the Gibbs measure comes arbitrarily close (in a metric which generates the weak topology) to a non-Gaussian measure which depends on the realization of the random patterns. This random measure is given explicitly by its (random) density.

Appeared in

  • Markov Process. Related Fields 5 (1999), no. 4. 423-449

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