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Decay to equilibrium in random spin systems on a lattice

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Abstract

We study continuous and discrete spin systems on a lattice with random interactions of finite range. In particular for sufficiently large interactions we prove no spectral gap property in the high temperature region. Moreover we show that in two dimensions, if the temperature is sufficiently high and the probability of interaction to be large is small enough, we have an almost sure stretched exponential upper bound for the decay to equilibrium.

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References

  1. Aizenman, M., Holley, R.: Rapid Convergence to Equilibrium of Stochastic Ising Models in the Dobrushin-Shlosman Regime. In: Percolation Theory and Ergodic Theory of Infinite Particle Systems, ed. Kesten, H., Berlin-Heidelberg-New York: Springer-Verlag 1987, pp. 1–11

    Google Scholar 

  2. Ben Arous, G., Guionnet, A.: Large Deviations for Langevin Spin Glass Dynamics. Probab. Theory Relat. Fields102, 455–509 (1995)

    Article  Google Scholar 

  3. Ben Arous, G., Guionnet, A.: Annealed and Quenched Propagation of Chaos for Asymmetric Spin Glass Dynamics. Submitted to Probab. Theory Relat. Fields

  4. Ben Arous, G. Guionnet, A.: Symmetric Langevin spin glass dynamics. Submitted to Annals of Probab.

  5. Beretti, A.: Some Properties of Random Ising models. J. Stat. Phys.38, 438–496 (1985)

    Google Scholar 

  6. Von Dreyfus, H., Klein A., Perez, J.F.: Taming Griffiths singularities: Infinite differentiability of quenched correlation functions. Commun. Math. Phys.170, 21–39 (1995)

    Google Scholar 

  7. Dobrushin, R.L., Bassalygo, L.A.: Uniqueness of a Gibbs Field with random potential — An elementary approach. Theory Probab. Appl.31, No. 4, 572–589 (1986)

    Article  Google Scholar 

  8. Dobrushin, R.L., Martirosyan, M.R.: Possibility of the High Temperature Phase Transitions Due to the Many-Particle Nature of the Potential. Theor. Math. Phys.75, 443–448 (1988)

    Article  Google Scholar 

  9. Dobrushin, R.L., Shlosman, S.: Constructive criterion for the uniqueness of Gibbs field. In: Statistical Physics and Dynamical Systems, Rigorous Results, Eds. Fritz, J., Jaffe, A., and Szasz, D., Basel-Boston: Birkhäuser, 1985, pp. 347–370

    Google Scholar 

  10. Dobrushin, R.L., Shlosman, S.: Completely analytical Gibbs fields. In: Statistical Physics and Dynamical Systems, Rigorous Results, Eds. Fritz, J., Jaffe, A., Szasz, D., Basel-Boston: Birkhäuser, 1985, pp. 371–403

    Google Scholar 

  11. Dobrushin, R.L., Shlosman, S.: Completely analytical interactions: Constructive description. J. Stat. Phys.46, 983–1014 (1987)

    Article  Google Scholar 

  12. Dunlop, F.: Correlation Inequalities for multicomponent Rotators. Commun. Math. Phys.49, 247–256 (1976)

    Google Scholar 

  13. Fröhlich, J.: Mathematical Aspects of the Physics of Disordered Systems. In: Critical Phenomena, Random Systems, Gauge Theories, Eds. Osterwalder, K., Stora, R., Amsterdam: Elsevier, 1986

    Google Scholar 

  14. Fröhlich, J., Zegarlinski, B.: The High-Temperature Phase of Long-Range Spin Glasses. Commun. Math. Phys.110, 121–155 (1987)

    Article  Google Scholar 

  15. Fröhlich, J., Zegarlinski, B.: Some Comments on the Sherrington-Kirkpatrick Model of Spin Glasses. Commun. Math. Phys.112, 553–566 (1987)

    Google Scholar 

  16. Fröhlich, J., Zegarlinski, B.: Spin Glasses and Other Lattice Systems with Long-Range Interactions. Commun. Math. Phys.120, 665–688 (1989)

    Article  Google Scholar 

  17. Van Enter, A.C.D., Zegarlinski, B.: A Remark on Differentiability of the Pressure Functional. Rev. Math. Phys.17, 959–977 (1995)

    Article  Google Scholar 

  18. Fröhlich, J., Imbrie, J.Z.: Improved Perturbation Expansion for Disordered Systems: Beating Griffiths Singularities. Commun. Math. Phys.96, 145–180 (1984)

    Article  Google Scholar 

  19. Fröhlich, J., Spencer, T.: The Kosterlitz-Thouless Transition in Two-Dimensional Abelian Spin Systems and the Coulomb Gas. Commun. Math. Phys.81, 527–602 (1981)

    Article  Google Scholar 

  20. Gelis, G., Maes, C.: Percolation Techniques in Disordered Spin Flip Dynamics: Relaxation to the Unique Invariant Measure. K.U. Leuven, Preprint 1995

  21. Griffiths, R.: Non-Analytic Behaviour Above the Critical Point in a Random Ising Ferromagnet. Phys. Rev. Lett.23, 17–19 (1969).

    Article  Google Scholar 

  22. Gross, L.: Logarithmic Sobolev inequalities. Am. J. Math.97, 1061–1083 (1976)

    Google Scholar 

  23. Guionnet, A: Thesis, Université Paris-Sud, March 1995

  24. Holley, R., Stroock, D.: Logarithmic Sobolev inequalities and stochastic Ising models. J. Stat. Phys.46, 1159–1194 (1987)

    Article  Google Scholar 

  25. Kesten, H.: Aspect of first passage percolation. Ecole d'ete de St Flour, LNM1180, pp. 125–264 (1986)

    Google Scholar 

  26. Lu Sheng Lin., Yau, Horng-Tzer: Spectral Gap and Logarithmic Sobolev Inequality for Kawasaki and Glauber Dynamics. Commun. Math. Phys.156, 399–433 (1993)

    Article  Google Scholar 

  27. Majewski, A.W., Zegarlinski, B.: Quantum Stochastic Dynamics I: Spin Systems on a Lattice. Math. Phys. Electr. J.1, Paper 2 (1995)

    Google Scholar 

  28. Martinelli, F., Olivieri, E.: Approach to Equilibrium of Glauber Dynamics in the One Phase Region: I. The Attractive case/II. The General Case. Commun. Math. Phys.161, 447–486 and 487–514 (1994)

    Google Scholar 

  29. Martinelli, F., Olivieri, E., Shonmann, R. H.: For 2-D Lattice Spin Systems Weak Mixing Implies Strong Mixing. Commun. Math. Phys.165, 33–47 (1994)

    Article  Google Scholar 

  30. Pfister, C.-E.: Translation Invariant Equilibrium States of Ferromagnetic Abelian Lattice Systems. Commun. Math. Phys.86, 375–390 (1982)

    Article  Google Scholar 

  31. Ruelle, D.: Statistical Mechanics: Rigorous Results. New York: W. A. Benjamin Inc., 1969

    Google Scholar 

  32. Schonmann, R.H., Schlosman, S.B.: Complete Analyticity for 2D Ising Completed. Preprint 1995

  33. Stroock, D.W., Zegarlinski, B.: The Logarithmic Sobolev inequality for Continuous Spin Systems on a Lattice. J. Funct. Anal.104, 299–326 (1992)

    Google Scholar 

  34. Stroock, D.W., Zegarlinski, B.: The Equivalence of the Logarithmic Sobolev Inequality and the Dobrushin-Shlosman Mixing Condition. Commun. Math. Phys.144, 303–323 (1992)

    Article  Google Scholar 

  35. Stroock, D.W., Zegarlinski, B.: The Logarithmic Sobolev inequality for Discrete Spin Systems on a Lattice. Commun. Math. Phys.149, 175–193 (1992)

    Article  Google Scholar 

  36. Stroock, D.W., Zegarlinski, B.: The ergodic properties of Glauber dynamics. J. Stat. Phys.81, 1007–1019 (1995)

    Google Scholar 

  37. Thomas, L.E.: Bound on the Mass Gap for Finite Volume Stochastic Ising Models at Low Temperature. Commun. Math. Phys.126, 1–11 (1989)

    Google Scholar 

  38. Zegarlinski, B.: Interactions and Pressure Functionals for Disordered Lattice Systems. Commun. Math. Phys.139, 305–339 (1991)

    Google Scholar 

  39. Zegarlinski, B.: Strong Decay to Equilibrium in One Dimensional Random Spin Systems. J. Stat. Phys.77, 717–732 (1994)

    Google Scholar 

  40. Zegarlinski, B.: Spin Systems with Long-Range Interactions. Rev. Math. Phys.6, 115–134 (1994)

    Article  Google Scholar 

  41. Zegarlinski, B.: Ergodicity of Markov Semigroups. In: Proc. of the Conference: Stochastic Partial Differential Equations, Edinburgh 1994, Ed. A. Etheridge, LMS Lecture Notes216, Cambridge, Cambridge University Press 1995, pp. 312–337

    Google Scholar 

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Authors and Affiliations

  1. Mathematiques, CNRS URA 0743, Universite Paris-Sud Orsay, 91405, France

    Alice Guionnet

  2. Department of Mathematics, Imperial College London, SW7 2BZ, UK

    Boguslaw Zegarlinski

Authors
  1. Alice Guionnet
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  2. Boguslaw Zegarlinski
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Additional information

Communicated by Ya. G. Sinai

Supported by EEC grant SC1-CT92-784.

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Guionnet, A., Zegarlinski, B. Decay to equilibrium in random spin systems on a lattice. Commun.Math. Phys. 181, 703–732 (1996). https://doi.org/10.1007/BF02101294

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  • DOI: https://doi.org/10.1007/BF02101294

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