Probability Theory and Related Fields

, Volume 102, Issue 4, pp 455–509

Large deviations for Langevin spin glass dynamics

  • G. B. Arous
  • A. Guionnet
Article
  • 195 Downloads

Summary

We study the asymptotic behaviour of asymmetrical spin glass dynamics in a Sherrington-Kirkpatrick model as proposed by Sompolinsky-Zippelius. We prove that the annealed law of the empirical measure on path space of these dynamics satisfy a large deviation principle in the high temperature regime. We study the rate function of this large deviation principle and prove that it achieves its minimum value at a unique probability measureQ which is not markovian. We deduce that the quenched law of the empirical measure converges to δQ. Extending then the preceeding results to replicated dynamics, we investigate the quenched behavior of a single spin. We get quenched convergence toQ in the case of a symmetric initial law and even potential for the free spin.

Mathematics Subject Classification

60F10 60H10 60K35 82C44 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Aizenman, M., Lebowitz, J.L., Ruelle, D.: Some rigorous results on the Sherrington-Kirkpatrick spin glass model. Commun. Math. Phys.112, 3–20 (1987)Google Scholar
  2. 2.
    Ben Arous, G., Brunaud, M.: Methode de Laplace: Etude variationnelle des fluctuations de diffusions de type “champ moyen”. Stochastics31–32, 79–144 (1990)Google Scholar
  3. 3.
    Comets, F., Neveu, J.: Modèle de Sherrington-Kirkpatrick à haute temperature et calcul stochastique. Preprint, Ecole PolytechniqueGoogle Scholar
  4. 4.
    Dawson, D.A., Gartner, J.: Large déviations from the McKean-Vlasov limit for weakly interacting diffusions. Stochastics20, 247–308 (1987)Google Scholar
  5. 5.
    Deuschel, J.-D., Stroock, D.W.: Large deviations. New York: Academic PressGoogle Scholar
  6. 6.
    Dobrushin: Prescribing a system of random variables with conditional distributions. Theoret. Probab. Appl.15, 458 (1970)Google Scholar
  7. 7.
    Jacod, J.: Calcul stochastique et Problèmes de martingales (Lect. Notes Math., vol. 714) Berlin: Springer 1979Google Scholar
  8. 8.
    Ledoux, M., Talagrand, M.: Probability in Banach spaces, Isoperimetry and Processes. Berlin: Springer 1991Google Scholar
  9. 9.
    Mezard, M., Parisi, G., Virasoro, M.: Spin glass theory and beyond. World Scientific Lecture Notes in Physics 1987Google Scholar
  10. 10.
    Neveu, J.: Processus aléatoires gaussiens. Presses de l'université de MontréalGoogle Scholar
  11. 11.
    Parisi, G.: A sequence of approximated solutions to the S-K model for spin glasses. J. Phys.A13, L115-L121 (1980)Google Scholar
  12. 12.
    Rachev, T.: Probability metrics and the stability of stochastics models. New York: WileyGoogle Scholar
  13. 13.
    Revuz, D., Yor M.: Continuous martingales and brownian motion. Berlin: Springer 1991Google Scholar
  14. 14.
    Sznitman, A.-S.: Non linear reflecting diffusion process and the propagation of chaos and fluctuations associated. J. Funct. Anal.56, 311–336 (1984)Google Scholar
  15. 15.
    Sznitman, A.-S.: Equation de type Boltzman spatialement homogenes. Z.f. Wahrscheinlichkeitstheorie verw. Gebiete,66, 559–592 (1994)Google Scholar
  16. 16.
    Sompolinsky, H., Zippelius, A.: Phys. Rev. Lett.47, 359 (1981)Google Scholar
  17. 17.
    Talagrand, M.: Concentration of measure and isoperimetric inequalities in product space. Preprint (1994)Google Scholar
  18. 18.
    Tanaka, H.: Limit theorems for certain diffusion processes. Proc. Taniguchi Symp, Katata 1982, 469–488, Tokyo, Kinokuniya (1984)Google Scholar

Copyright information

© Springer-Verlag 1995

Authors and Affiliations

  • G. B. Arous
    • 1
  • A. Guionnet
    • 2
  1. 1.URA 762, CNRS, DMIEcole Normale SuperieureParisFrance
  2. 2.URA 743, CNRSUniversité de Paris SudOrsayFrance

Personalised recommendations