WIAS Preprint No. 79, (1994)

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



Authors

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

2010 Mathematics Subject Classification

  • 60K35 82B44 82C32

Keywords

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

DOI

10.20347/WIAS.PREPRINT.79

Abstract

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

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