Journal of Statistical Physics

, Volume 84, Issue 5–6, pp 973–986 | Cite as

Dynamics in the Sherrington-Kirkpatrick model. I. The first step

  • A. E. Patrick


We study properties of the random configuration {s j (1)} j=1 N produced by the first step of the parallel dynamics in the Sherrington-Kirkpatrick model. We show that the law of large numbers holds for the sequence of overlaps between the initial (nonrandom) configuration {s j (0)} j=1 N and {s j (1)} j=1 N , and obtain the distribution of the fluctuations around the limiting value. As a by-product we derive the average number of the fixed points {s j (1)} j=1 N with a given value of the magnetization\(m_N = (1/N)\sum\nolimits_{j = 1}^N {s_j (0)} \).

Key Words

Spin-glass dynamics fixed points fluctuations 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    J. M. G. Amaro de Matos, A. E. Patrick, and V. A. Zagrebnov, Random infinite-volume Gibbs states for the Curie-Weiss random field Ising model,J. Stat. Phys. 66:139–164 (1992).Google Scholar
  2. 2.
    D. C. Dean, On the metastable states of the zero-temperature SK model,J. Phys. A: Math. Gen. 27:L899-L905 (1994).Google Scholar
  3. 3.
    E. Gardner, Structure of metastable states in the Hopfield model,J. Phys. A: Math. Gen. 19:L1047-L1052 (1986).Google Scholar
  4. 4.
    E. Gardner, B. Derrida, and P. Mottishaw, Zero temperature parallel dynamics for infinite range spin glasses and neural networks,J. Phys. (Paris)48:741–755 (1987).Google Scholar
  5. 5.
    S. Kirkpatrick and D. Sherrington, Infinite-ranged models of spin glasses,Phys. Rev. B 17:4384–4403 1978).Google Scholar
  6. 6.
    C. M. Newman, Memory capacity in neural networks model: Rigorous lower bounds,Neural Networks 1:223–238 (1988).Google Scholar
  7. 7.
    F. Tanaka and S. F. Edwards,J. Phys. F 10:2471 (1980).Google Scholar
  8. 8.
    A. Treves and D. J. Amit, Metastable states in asymmetrically diluted Hopfield networks,J. Phys. A: Math. Gen. 21:3155–3169 (1988).Google Scholar

Copyright information

© Plenum Publishing Corporation 1996

Authors and Affiliations

  • A. E. Patrick
    • 1
  1. 1.Centre de Physique Théorique, CNRSMarseille Cedex 9France

Personalised recommendations