Abstract
An interacting particle system (Glauber dynamics) which evolves on a finite subset in thed-dimensional integer lattice is considered. It is known that a mixing property of the Gibbs state in the sense of Dobrushin and Shlosman is equivalent to several very strong estimates in terms of the Glauber dynamics. We show that similar, but seemingly much milder estimates are again equivalent to the Dobrushin-Shlosman mixing condition, hence to the original ones found by Stroock and Zegarlinski. This may be understood as the absence of intermediate speed of convergence to equilibrium.
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References
M. Aizenmann and R. Holley, Rapid convergence to equilibrium of stochastic Ising models in the Dobrushin-Shlosman regime, inPercolation Theory and Infinite Particle Systems, H. Kesten, ed. (Springer, Berlin, 1987), pp. 1–11.
J. D. Deuschel and D. W. Stroock,Large Deviations (Academic Press, New York, 1989).
R. L. Dobrushin and S. Shlosman, Completely analytical Gibbs fields, inStatistical Physics and Dynamical Systems, J. Fritz, A. Jaffe, and D. Szasz, eds. (Birkhäuser, Basel, 1985).
R. L. Dobrushin and S. Shlosman, Completely analytical interactions: Constructive description,J. Stat. Phys. 46:983–1014 (1987).
Y. Higuchi, Private communication.
Y. Higuchi and N. Yoshida, Slow relaxation of stochastic Ising models with random and non-random boundary conditions, inNew Trends in Stochastic Analysis, K. D. Elworthy, S. Kusuoka, and I. Shigekawa, eds. (World Scientific Publishing, 1997).
R. Holley, Possible rate of convergence in finite range, attractive spin systems,Contemp. Math. 41:215–234 (1985).
R. Holley and D. W. Stroock, Logarithmic Sobolev inequality and-stochastic Ising models,J. Stat. Phys. 46:1159–1194 (1987).
E. Laroche, Hypercontractivité pour des systèm de spins de portée infinie,Prob. Theory Related Fields 101:89–132 (1995).
T. M. Liggett,Interacting Particle Systems (Springer-Verlag, Berlin, 1985).
F. Martinelli and E. Olivieri, Approach to equilibrium of Glauber dynamics in the one phase region I: Attractive case,Commun. Math. Phys. 161:447–486 (1994).
F. Martinelli and E. Olivieri Approach to equilibrium of Glauber dynamics in the one phase region II: General case,Commun. Math. Phys. 161:487–514 (1994).
F. Martinelli, E. Olivieri, and R. Schonmann, For 2-D lattice spin systems weak mixing implies strong mixing,Commun. Math. Phys. 165:33–47 (1994).
R. H. Schonmann and S. B. Schlosman, Complete analyticity for 2D Ising completed,Commun. Math. Phys. 170:453–482 (1995).
D. W. Stroock and B. Zegarlinski, The equivalence of the logarithmic Sobolev inequality and the Dobrushin-Shlosman mixing condition,Commun. Math. Phys. 144:303–323 (1992).
D. W. Stroock and B. Zegarlinski, The logarithmic Sobolev inequality for discrete spin systems on the lattice,Commun. Math. Phys. 149:175–193 (1992).
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Yoshida, N. Relaxed criteria of the Dobrushin-Shlosman mixing condition. J Stat Phys 87, 293–309 (1997). https://doi.org/10.1007/BF02181489
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DOI: https://doi.org/10.1007/BF02181489