Probability Theory and Related Fields

, Volume 136, Issue 4, pp 619–660 | Cite as

Cugliandolo-Kurchan equations for dynamics of Spin-Glasses



We study the Langevin dynamics for the family of spherical p-spin disordered mean-field models of statistical physics. We prove that in the limit of system size N approaching infinity, the empirical state correlation and integrated response functions for these N-dimensional coupled diffusions converge almost surely and uniformly in time, to the non-random unique strong solution of a pair of explicit non-linear integro-differential equations intensively studied by Cugliandolo and Kurchan.

Mathematics Subject Classification

Primary: 82C44 Secondary: 82C31 60H10 60F15 60K35 

Key words or phrases

Interacting random processes Disordered systems Statistical mechanics Langevin dynamics Aging p-spin models 


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  1. 1.
    Aida, S., Stroock, D.: Moment estimates derived from Poincaré and logarithmic Sobolev inequalities. Math. Res. Lett. 1, 75–86 (1994)MATHMathSciNetGoogle Scholar
  2. 2.
    Ané, C. et altri: Sur les inégalités de Sobolev logarithmiques. Panoramas et Syntheses, 10, Société Mathématique de France (2000).Google Scholar
  3. 3.
    Ben Arous, G.: Aging and spin-glass dynamics. Proceedings of the International Congress of Mathematicians, Vol. III , 3–14, Higher Ed. Press, Beijing, 2002Google Scholar
  4. 4.
    Ben Arous, G., Bovier, A., Gayrard, V.: Glauber dynamics of the random energy model. II. Aging below the critical temperature. Comm. Math. Phys. 236, 1–54 (2003)MATHGoogle Scholar
  5. 5.
    Ben Arous, G., Dembo, A., Guionnet, A.: Aging of spherical spin glasses, Prob. Th. Rel. Fields. 120, 1–67 (2001)MATHCrossRefGoogle Scholar
  6. 6.
    Ben Arous, G., Guionnet, A.: Large deviations for Langevin spin glass dynamics. Prob. Th. Rel. Fields 102, 455–509 (1995)MATHMathSciNetCrossRefGoogle Scholar
  7. 7.
    Ben Arous, G., Guionnet, A.: Symmetric Langevin spin glass dynamics. Ann. Probab. 25, 1367–1422 (1997)MATHMathSciNetCrossRefGoogle Scholar
  8. 8.
    Bouchaud, J.P., Cugliandolo, L.F., Kurchan, J., Mezard, M.: Out of equilibrium dynamics in spin-glasses and other glassy systems. In: Young, A.P. (ed.) Spin glass dynamics and Random Fields 1997Google Scholar
  9. 9.
    Bovier, A., Gayrard, V.: The retrieval phase of the Hopfield model: a rigorous analysis of the overlap distribution. Probab. Theory Related Fields. 107, 61–98 (1997)MATHMathSciNetCrossRefGoogle Scholar
  10. 10.
    Crisanti, H., Horner, H., Sommers, H.-J.: The Spherical p-spin Interaction Spin Glass Model: The Dynamics, Zeitschrift fur Physik B 92, 257–271 (1993)Google Scholar
  11. 11.
    Cugliandolo, L.F.: Dynamics of glassy systems. In: Barrat, J-L. et al. (eds.) Slow relaxations and non-equilibirum dynamics in condensed matter. Springer, Berlin, 2003Google Scholar
  12. 12.
    Cugliandolo, L.F., Dean, D.S.: Full dynamical solution for a spherical spin-glass model. J. Phys. A: Math. Gen. 28, 4213–4234 (1995)MATHMathSciNetCrossRefGoogle Scholar
  13. 13.
    Cugliandolo, L.F., Kurchan, J.: Analytical solution of the off-equilibrium Dynamics of a Long-Range Spin-Glass Model. Phys. Rev. Lett. 71, 173 (1993)CrossRefGoogle Scholar
  14. 14.
    Dudley, R.: The sizes of compact subsets of Hilbert space and continuity of Gaussian Processes. J. Funct. Analysis 1, 290–330 (1967)MATHMathSciNetCrossRefGoogle Scholar
  15. 15.
    Grunwald, M.: Sanov results for Glauber spin-glass dynamics. Prob. Th. Rel. Fields 106, 187–232 (1996)MATHMathSciNetCrossRefGoogle Scholar
  16. 16.
    Guionnet, A.: Annealed and quenched propagation of chaos for Langevin spin glass dynamics. Prob. Th. Rel. Fields. 109, 183–215 (1997)MATHMathSciNetCrossRefGoogle Scholar
  17. 17.
    Guionnet, A., Mazza, C.: Long time behaviour of non-commutative processes solution of a linear differential equation. Prob. Theory. Rel. Fields. 131, 493–518 (2005)MATHMathSciNetCrossRefGoogle Scholar
  18. 18.
    Karatzas, I., Shreve, S.E.: Brownian motion and stochastic calculus, 2nd edn. Springer-Verlag, 1991Google Scholar
  19. 19.
    Ledoux, M., Talagrand, M.: Probability in Banach Spaces. Ergebnisse der Mathematik 23, Springer Verlag, 1991Google Scholar
  20. 20.
    Mezard, M., Parisi, G.: Virasoro, M. ; Spin glass theory and beyond. World Scientific Lecture Notes in Physic, 1987Google Scholar
  21. 21.
    Sompolinsky, H., Zippelius, A.: Relaxational dynamics of the Edwards-Anderson model and the mean-field theory of spin glasses. Phys. Rev. B. 25, 6860–6875 (1982)CrossRefGoogle Scholar
  22. 22.
    Talagrand, M.: Spin Glasses: a Challenge for Mathematicians. Ergebnisse der Mathematik 46, Springer Verlag, 2003Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2006

Authors and Affiliations

  • Gérard Ben Arous
    • 1
    • 2
  • Amir Dembo
    • 3
  • Alice Guionnet
    • 4
  1. 1.Courant Institute of Mathematical SciencesNew YorkUSA
  2. 2.EPFLLausanneSwitzerland
  3. 3.Department of Statistics and Department of MathematicsStanford UniversityStanfordUSA
  4. 4.UMPA, Ecole Normale Superieure de Lyon 46 allée d'ItalieLyon Cedex 07France

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