Abstract
We consider the two-dimensional stochastic Ising model in finite square Λ with free boundary conditions, at inverse temperature β>β0 and zero external field. Using duality and recent results of Ioffe on the Wulff construction close to the critical temperature, we extend some of the results obtained by Martinelli in the low-temperature regime to any temperature below the critical one. In particular we show that the gap in the spectrum of the generator of the dynamics goes to zero in the thermodynamic limit as an exponential of the side length of Λ, with a rate constant determined by the surface tension along one of the coordinate axes. We also extend to the same range of temperatures the result due to Shlosman on the equilibrium large deviations of the magnetization with free boundary conditions.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
J. Chayes, L. Chayes, and R. H. Schonmann, Exponential decay of connectivity in the two-dimensional Ising model,J. Stat. Phys. 49:433 (1987).
R. Dobrushin, R. Kotecký, and S. Shlosman,Wulff Construction. A Global Shape from Local Interaction (American Mathematical Society, Providence, Rhode Island, 1992).
R. L. Dobrushin and S. Shlosman, Constructive criterion for the uniqueness of Gibbs fields, inStatistical Physics and Dynamical Systems, Fritz, Jaffe, and Szász, eds. (Birkhauser, Boston, 1985), p. 347.
C. M. Fortuin, P. W. Kasteleyn, and J. Ginibre, Correlation inequalities on some partially ordered sets,Commun. Math. Phys. 22:89 (1971).
Y. Higuchi, Coexistence of infinite (*)-cluster II: Ising percolation in two dimension,Prob. Theory Related Fields 97:1 (1993).
Y. Higuchi and N. Yoshida, Slow relaxation of 2D-stochastic Ising models at low temperature, preprint (1995).
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 toT c for the Ising model in two dimensions, preprint (1994).
M. Jerrum and A. Sinclair, Polynomial-time approximation algorithms for the Ising model,SIAM J. Comput. 22:1087 (1993).
T. M. Ligget,Interacting Particles Systems (Springer-Verlag, Berlin, 1985).
F. Martinelli, On the two dimensional dynamical Ising model in the phase coexistence region,J. Stat. Phys. 76(5/6):1179 (1994).
E. Marcelli and F. Martinelli, Some new results on the 2d kinetic Ising model in the phase coexistence region, preprint (1995).
F. Martinelli and E. Olivieri, Approach to equilibrium of Glauber dynamics in the one phase region I: The attractive case,Commun. Math. Phys. 161:447 (1994).
F. Martinelli, E. Olivieri, and R. H. Schonmann, For Gibbs state of 2D lattice spin systems weak mixing implies strong mixing,Commun. Math. Phys. 165:33 (1994).
F. Martinelli, E. Olivieri, and E. Scoppola, Metastability and exponential approach to equilibrium for low temperature stochastic Ising model,J. Stat. Phys. 61:1105 (1990).
C. E. Pfister, Large deviations and phase separation in the two-dimensional Ising model,Helv. Phys. Acta 64:953 (1991).
S. B. Shlosman, The droplet in the tube: A case of phase transition in the canonical ensemble,Commun. Math. Phys. 125:81 (1989).
B. Simon, Correlation inequalities and the decay of correlations in ferromagnets,Commun. Math. Phys. 77:111 (1980).
R. H. Schonmann and S. B. Shlosman, Complete analiticity for 2D Ising model completed,Commun. Math. Phys. 170:453 (1995).
R. H. Schonmann and S. B. Shlosman, Constrained variational problems with applications to the Ising model, preprint.
R. H. Schonmann and S. B. Shlosman, In preparation.
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Cesi, F., Guadagni, G., Martinelli, F. et al. On the two-dimensional stochastic Ising model in the phase coexistence region near the critical point. J Stat Phys 85, 55–102 (1996). https://doi.org/10.1007/BF02175556
Received:
Issue Date:
DOI: https://doi.org/10.1007/BF02175556