WIAS Preprint No. 79, (1994)

Gibbs states of the Hopfield model in the regime of perfect memory


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

2010 Mathematics Subject Classification

  • 60K35 82B44 82C32


  • thermodynamic properties of the Hopfield model, infinite number of infinite volume Gibbs measures, Bernoulli measures, convergent sequences of measures




We study the thermodynamic properties of the Hopfield model of an autoassociative memory. If N denotes the number of neurons and M(N) the number of stored patterns, we prove the following results: If M ⁄N ↓ 0 as N ↑ ∞, then there exists an infinite number of infinite volume Gibbs measures for all temperatures T < 1 concentrated on spin configurations that have overlap with exactly one specific pattern. Moreover, the measures induced on the overlap parameters are Dirac measures concentrated on a single point. If M ⁄N → α, as N ↑ ∞ for a small enough, we show that for temperatures T smaller than some T(α) < 1, the induced measures can have support only on a disjoint union of balls around the previous points, but we cannot construct the infinite volume measures through convergent sequences of measures.

Appeared in

  • Probab. Theor. Relat. Fields, 100 (1994), pp. 329--363

Download Documents