Skip to main content

Large deviations for Langevin spin glass dynamics

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 δ Q . 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.

References

  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. 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. Comets, F., Neveu, J.: Modèle de Sherrington-Kirkpatrick à haute temperature et calcul stochastique. Preprint, Ecole Polytechnique

  4. Dawson, D.A., Gartner, J.: Large déviations from the McKean-Vlasov limit for weakly interacting diffusions. Stochastics20, 247–308 (1987)

    Google Scholar 

  5. Deuschel, J.-D., Stroock, D.W.: Large deviations. New York: Academic Press

  6. Dobrushin: Prescribing a system of random variables with conditional distributions. Theoret. Probab. Appl.15, 458 (1970)

    Google Scholar 

  7. Jacod, J.: Calcul stochastique et Problèmes de martingales (Lect. Notes Math., vol. 714) Berlin: Springer 1979

    Google Scholar 

  8. Ledoux, M., Talagrand, M.: Probability in Banach spaces, Isoperimetry and Processes. Berlin: Springer 1991

    Google Scholar 

  9. Mezard, M., Parisi, G., Virasoro, M.: Spin glass theory and beyond. World Scientific Lecture Notes in Physics 1987

  10. Neveu, J.: Processus aléatoires gaussiens. Presses de l'université de Montréal

  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. Rachev, T.: Probability metrics and the stability of stochastics models. New York: Wiley

  13. Revuz, D., Yor M.: Continuous martingales and brownian motion. Berlin: Springer 1991

    Google Scholar 

  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. Sznitman, A.-S.: Equation de type Boltzman spatialement homogenes. Z.f. Wahrscheinlichkeitstheorie verw. Gebiete,66, 559–592 (1994)

    Google Scholar 

  16. Sompolinsky, H., Zippelius, A.: Phys. Rev. Lett.47, 359 (1981)

    Google Scholar 

  17. Talagrand, M.: Concentration of measure and isoperimetric inequalities in product space. Preprint (1994)

  18. Tanaka, H.: Limit theorems for certain diffusion processes. Proc. Taniguchi Symp, Katata 1982, 469–488, Tokyo, Kinokuniya (1984)

    Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Rights and permissions

Reprints and Permissions

About this article

Cite this article

Arous, G.B., Guionnet, A. Large deviations for Langevin spin glass dynamics. Probab. Th. Rel. Fields 102, 455–509 (1995). https://doi.org/10.1007/BF01198846

Download citation

  • Received:

  • Revised:

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF01198846

Mathematics Subject Classification

  • 60F10
  • 60H10
  • 60K35
  • 82C44