WIAS Preprint No. 97, (1994)

Gibbs states of the Hopfield model with extensively many patterns


  • Bovier, Anton
  • Gayrard, Véronique
  • Picco, Pierre

2010 Mathematics Subject Classification

  • 82B44 82C32


  • Hopfield model, Gibbs states, self-averaging, spin glasses




We consider the Hopfield model with M(N) = αN patterns, where N is the number of neurons. We show that if α is sufficiently small and the temperature sufficiently low, then there exist disjoint Gibbs states for each of the stored patterns, almost surely with respect to the distribution of the random patterns. This solves a problem left open in previous work [BGPl]. The key new ingredient is a self averaging result on the free energy functional. This result has considerable additional interest and some consequences are discussed. A similar result for the free energy of the Sherrington-Kirkpatrick model is also given.

Appeared in

  • J. Statist. Phys., 79 (1995), pp. 395--414

Download Documents