Abstract
We consider Glauber-type dynamics for disordered Ising spin systems with nearest neighbor pair interactions in the Griffiths phase. We prove that in a nontrivial portion of the Griffiths phase the system has exponentially decaying correlations of distant functions with probability exponentially close to 1. This condition has, in turn, been shown elsewhere to imply that the convergence to equilibrium is faster than any stretched exponential, and that the average over the disorder of the time-autocorrelation function goes to equilibrium faster than exp[−k(log t)d/(d−1)]. We then show that for the diluted Ising model these upper bounds are optimal.
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REFERENCES
M, Aizenman and D. J. Barsky, Sharpness of the phase transition in percolation models, Commun. Math. Phys. 108:489–256 (1987).
M. Aizenman, D. J. Barsky, and R. Fernandez, The phase transition in a general class of Ising-type models is sharp, J. Stat. Phys. 47(3/4):343 (1987).
K. S. Alexander and L. Chayes, Non-perturbative criteria for Gibbsian uniqueness, Commun. Math. Phys. 189:447–464 (1997).
M. Aizenman, J. T. Chayes, L. Chayes, and C. M. Newman, The phase boundary in dilute and random Ising and Potts ferromagnets, J. Phys. A. Math. Gen. 20:L313 (1987).
J. Chayes, L. Chayes, and R. H. Schonmann, Exponential decay of connectivity in the two-dimensional Ising model, J. Stat. Phys. 49:433 (1987).
F. Cesi, G. Guadagni, F. Martinelli, and R. H. Schonmann, On the two-dimensional stochastic Ising model in the phase coexistence region near the critical point, J. Stat. Phys. 85(1/2):55–102 (1996).
F. Cesi, C. Maes, and F. Martinelli, Relaxation of disordered magnets in the Griffiths regime, Commun. Math. Phys. 188:135–173 (1997).
F. Cesi, C. Maes, and F. Martinelli, Relaxation to equilibrium for two dimensional disordered Ising systems in the Griffiths phase, Commun. Math. Phys. 189:323–335 (1997).
R. Dobrushin, R. Kotecký, and S. Shlosman, Wulff construction. A global shape from local interaction, Translation of Mathematical Monographs, Vol. 104, AMS, 1992.
C. M. Fortuin and P. W. Kasteleyn, On the random cluster model. I. Introduction and relation to other models, Physica 57:536–564 (1972).
R. Griffiths, Non-analytic behaviour above the critical point in a random Ising ferromagnet, Phys. Rev. Lett. 23:17 (1969).
G. R. Grimmett and J. M. Marstrand, The supercritical phase of percolation is well behaved, Proc. Royal Soc. (London), Ser. A 430:439–457 (1990).
Y. Higuchi, Coexistence of infinite (*)-clusters II: Ising percolation in two dimension, Probab. Theory Rel. Fields 97:1 (1993).
D. Ioffe, Large deviation for the 2D Ising model: a lower bound without cluster expansion, J. Stat. Phys. 74(1/2):411 (1994).
D. Ioffe, Exact large deviation bounds up to T c for the Ising model in two dimension, Probab. Theory Related Fields 102:313–330 (1995).
D. Ioffe and R. H. Schonmann, Dobrushin-Kotecky-Shlosman theorem up to the critical temperature. Preprint (1997).
M. V. Menshikov, S. A. Molchanov, and S. A. Sidovenko, Percolation theory and some applications, Itogi Nauki i Techniki (Series of Probability Theory, Mathematical Statistics and Theoretical Cybernetics) 24:53–110 (1986).
F. Martinelli and E. Olivieri, Approach to equilibrium of Glauber dynamics in the one phase region I: The attractive case, Commun. Math. Phys. 161 (1994).
E. Olivieri, F. Perez, and S. Goulart-Rosa, Jr., Some rigorous results on the phase diagram of the dilute Ising model, Phys. Lett. 94A(6/7):309 (1983).
A. Pisztora, Surface order large deviations for Ising, Potts and percolation models, Probab. Theory Rel. Fields 104:427 (1996).
C. E. Pfister, Large deviations and phase separation in the two-dimensional Ising model, Helvetica Physica Acta 64:953 (1991).
R. H. Schonmann, Second order large deviation estimates for ferromagnetic systems in the phase coexistence region, Commun. Math. Phys. 112:409 (1987).
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Alexander, K.S., Cesi, F., Chayes, L. et al. Convergence to Equilibrium of Random Ising Models in the Griffiths Phase. Journal of Statistical Physics 92, 337–351 (1998). https://doi.org/10.1023/A:1023077101354
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DOI: https://doi.org/10.1023/A:1023077101354