Probability Theory and Related Fields

, Volume 107, Issue 1, pp 61–98 | Cite as

The retrieval phase of the Hopfield model: A rigorous analysis of the overlap distribution*

  • Anton Bovier
  • Véronique Gayrard

Summary.

Standard large deviation estimates or the use of the Hubbard–Stratonovich transformation reduce the analysis of the distribution of the overlap parameters essentially to that of an explicitly known random function ΦN,β on M. In this article we present a rather careful study of the structure of the minima of this random function related to the retrieval of the stored patterns. We denote by m* (β ) the modulus of the spontaneous magnetization in the Curie–Weiss model and by α the ratio between the number of the stored patterns and the system size. We show that there exist strictly positive numbers 0 < γa < γc such that (1) If √α≦γa (m* (β ))2, then the absolute minima of Φ are located within small balls around the points ± m* eμ , where eμ denotes the μ-th unit vector while (2) if √α≦γc (m* (β ))2 at least a local minimum surrounded by extensive energy barriers exists near these points. The random location of these minima is given within precise bounds. These are used to prove sharp estimates on the support of the Gibbs measures.

Mathematics Subject Classification (1991): 82B44 60K35 82C32 

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Copyright information

© Springer-Verlag Berlin Heidelberg 1997

Authors and Affiliations

  • Anton Bovier
    • 1
  • Véronique Gayrard
    • 2
  1. 1.Weierstraß-Institut für Angewandte Analysis und Stochastik, Mohrenstraße 39, D-10117 Berlin, Germany (e-mail: bovier@ wias-berlin.de)DE
  2. 2.Centre de Physique Théorique - CNRS, Luminy, Case 907, F-13288 Marseille Cedex 9, France (e-mail: gayrard@ cpt.univ-mrs.fr)FR

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