The Annals of Probability

Symmetric Langevin spin glass dynamics

G. Ben Arous and A. Guionnet

Full-text: Open access

Abstract

We study the asymptotic behavior of symmetric spin glass dynamics in the Sherrington–Kirkpatrick model as proposed by Sompolinsky–Zippelius. We prove that the averaged law of the empirical measure on the path space of these dynamics satisfies a large deviation upper bound in the high temperature regime. We study the rate function which governs this large deviation upper bound and prove that it achieves its minimum value at a unique probability measure $Q$ which is not Markovian. We deduce an averaged and a quenched law of large numbers. We then study the evolution of the Gibbs measure of a spin glass under Sompolinsky–Zippelius dynamics. We also prove a large deviation upper bound for the law of the empirical measure and describe the asymptotic behavior of a spin on path space under this dynamic in the high temperature regime.

Article information

Source
Ann. Probab. Volume 25, Number 3 (1997), 1367-1422.

Dates
First available in Project Euclid: 18 June 2002

Permanent link to this document
http://projecteuclid.org/euclid.aop/1024404517

Mathematical Reviews number (MathSciNet)
MR1457623

Digital Object Identifier
doi:10.1214/aop/1024404517

Zentralblatt MATH identifier
0954.60031

Subjects
Primary: 60F10: Large deviations 60H10: Stochastic ordinary differential equations [See also 34F05] 60K35: Interacting random processes; statistical mechanics type models; percolation theory [See also 82B43, 82C43] 82C44: Dynamics of disordered systems (random Ising systems, etc.) 82C31: Stochastic methods (Fokker-Planck, Langevin, etc.) [See also 60H10] 82C22: Interacting particle systems [See also 60K35]

Keywords
Large deviations interacting random processes statistical mechanics Langevin dynamics

Citation

Ben Arous, G.; Guionnet, A. Symmetric Langevin spin glass dynamics. Ann. Probab. 25 (1997), no. 3, 1367--1422. doi:10.1214/aop/1024404517. http://projecteuclid.org/euclid.aop/1024404517.


Export citation

References

  • [1] Aizenman, M., Lebowitz, J. L. and Ruelle, D. (1987). Some rigorous results on the Sherrington-Kirkpatrick spin glass model. Comm. Math. Phys. 112 3-20.
  • [2] Ben Arous, G. and Brunaud, M. (1990). Methode de Laplace: ´etude variationnelle des fluctuations de diffusions de type "champ moyen." Stochastics 31-32 79-144.
  • [3] Ben Arous, G. and Guionnet, A. (1995). Large deviations for Langevin spin glass dynamics. Probab. Theory Related Fields 102 455-509.
  • [4] Ben Arous, G. and Guionnet, A. (1997). Langevin dynamics for Sherrington-Kirkpatrick spin glasses. Proceedings of a Conference on Spin Glasses, Berlin 1996.
  • [5] Bovier, A., Gayrard, V. and Picco, P. (1995). Gibbs states of the Hopfield model with extensively many patterns. J. Statist. Phys. 79 395-414.
  • [6] Comets, F. and Neveu, J. (1995). The Sherrington-Kirkpatrick model of spin glass and stochastic calculus: the high temperature case. Comm. Math. Phys. 166 549-564.
  • [7] Dawson, D. A. and Gartner, J. (1987). Large deviations from the McKean-Vlasov limit for weakly interacting diffusions. Stochastics 20 247-308.
  • [8] Deuschel, J-D. and Stroock, D. W. (1989). Large Deviations. Academic Press, New York.
  • [9] Fr ¨ohlich, J. and Zegarlinski, B. (1987). Some comments on the Sherrington-Kirkpatrick model of spin glasses. Comm. Math. Phys. 112 553-566.
  • [10] Guionnet, A. (1995). Dynamique de Langevin d'un verre de spins. Ph.D dissertation, Univ. Paris Sud, n. 3616.
  • [11] Guionnet, A. (1997). Averaged and quenched propagation of chaos for Langevin spin glass dynamics. Probab. Theory Related Fields. To appear.
  • [12] Mezard, M., Parisi, G. and Virasoro, M. (1987). Spin glass theory and beyond. Lecture Notes Phys. 9. World Scientific, Teaneck, NJ.
  • [13] Neveu, J. (1968). Processus Al´eatoires Gaussiens. Presses de l'universit´e de Montr´eal.
  • [14] Revuz, D. and Yor, M. (1991). Continuous Martingales and Brownian Motion. Springer, New York.
  • [15] Sompolinsky, H. and Zippelius, A. (1981). Phys. Rev. Lett. 47 359.
  • [16] Sznitman, A.-S. (1984). ´Equation de type de Boltzmann, spatialement homog enes. Z. Wahrsch. Verw. Gebiete 66 559-592.
  • [17] Talagrand, M. (1995). Concentration of measure and isoperimetric inequalities in product space. Inst. Hautes Etudes Sci. Publ. Math. 81 73-205.