Skip to main content
Log in

Surface Tension and the Ornstein–Zernike Behaviour for the 2D Blume–Capel Model

  • Published:
Journal of Statistical Physics Aims and scope Submit manuscript

Abstract

We prove existence of the surface tension in the low temperature 2D Blume–Capel model and verify the Ornstein–Zernike asymptotics of the corresponding finite-volume interface partition function.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Similar content being viewed by others

REFERENCES

  1. R. L. Dobrushin, Gibbs state describing coexistence of phases for a three-dimensional Ising model, Teor. Ver. Pril. 17:619-639 (1972); English transl. in Theor. Probability Appl. 17:582-600 (1972).

    Google Scholar 

  2. P. Holický, R. Kotecký, and M. Zahradník, Rigid interfaces for lattice models at low temperatures, J. Statist. Phys. 50:755-812 (1988).

    Google Scholar 

  3. S. Pirogov and Ya. Sinai, Phase diagrams of classical lattice systems, Theoret. and Math. Phys. 25:1185-1192 (1975) and 26:39-49 (1976).

    Google Scholar 

  4. P. Holický, R. Kotecký, and M. Zahradník, Phase diagram of horizontally invariant Gibbs states for lattice models, Ann. Henri Poincaré, in press.

  5. G. Gallavotti, The phase separation line in the two-dimensional Ising model, Commun. Math. Phys. 27:103-136 (1972).

    Google Scholar 

  6. M. Campanino, J. T. Chayes, and L. Chayes, Gaussian fluctuations of connectivities in the subcritical regime of percolation, Probab. Theory Related Fields 88:269-341 (1991).

    Google Scholar 

  7. M. Campanino and D. Ioffe, Ornstein-Zernike theory for the Bernoulli bond percolation on Zd, preprint (1999).

  8. J. T. Chayes and L. Chayes, Ornstein-Zernike behaviour for self-avoiding walks at all non-critical temperatures, Commun. Math. Phys. 105:221-238 (1986).

    Google Scholar 

  9. D. Ioffe, Ornstein-Zernike behaviour and analyticity of shapes for self-avoiding walks on Zd, Markov Process. Related Fields 4:323-350 (1998).

    Google Scholar 

  10. R. L. Dobrushin, R. Kotecký, and S. B. Shlosman, Wulff Construction: A Global Shape from Local Interaction. (Translations of Mathematical Monographs, Vol. 104) (Amer. Math. Soc., 1992).

  11. J. Bricmont and J. Slawny, Phase transitions in systems with a finite number of dominant ground states, J. Statist. Phys. 54:89-161 (1989).

    Google Scholar 

  12. J. Bricmont and J. Lebowitz, Wetting in Potts and Blume-Capel models, J. Statist. Phys. 46:1015-1029 (1987).

    Google Scholar 

  13. R. L. Dobrushin, A statistical behaviour of shapes of boundaries of phases, in Phase Transitions: Mathematics, Physics, Biology..., R. Kotecký, ed. (World Scientific, Singapore, 1993), pp. 60-70.

    Google Scholar 

  14. R. L. Dobrushin and O. Hryniv, Fluctuations of the phase boundary in the 2D Ising ferromagnet, Commun. Math. Phys. 189:395-445 (1997).

    Google Scholar 

  15. O. Hryniv, On local behaviour of the phase separation line in the 2D Ising model, Probab. Theory Related Fields 110:91-107 (1998).

    Google Scholar 

  16. R. Kotecký and D. Preiss, Cluster expansions for abstract polymer models, Commun. Math. Phys. 103:491-498 (1986).

    Google Scholar 

  17. D. B. Abraham and P. Reed, Interface profile of the Ising ferromagnet in two dimensions, Commun. Math. Phys. 49:35-46 (1976).

    Google Scholar 

  18. Y. Higuchi, On some limit theorems related to the phase separation line in the twodimensional Ising model, Z. Wahrsch. Verw. Gebiete 50:287-315 (1979).

    Google Scholar 

  19. J. Bricmont, J. L. Lebowitz, and C. E. Pfister, On the local structure of the phase separation line in the two-dimensional Ising system, J. Statist. Phys. 26:313-332 (1981).

    Google Scholar 

  20. E. Seneta, Non-negative Matrices and Markov Chains (Springer, 1981).

  21. W. Feller, An Introduction to Probability Theory and Its Applications, Vol. 1, 3rd ed. (Wiley, 1970).

  22. N. G. de Bruijn, Asymptotic Methods in Analysis, Bibliotheca Mathematica, Vol. IV (North-Holland, 1970).

  23. M. V. Fedoriuk, Asymptotics: Integrals and Series (Nauka, Moscow, 1987).

    Google Scholar 

  24. R. L. Dobrushin and S. B. Shlosman, Large and moderate deviations in the Ising model, in Probability Contributions to Statistical Mechanics, pp. 91-219, Adv. Soviet Math., Vol. 20 (Amer. Math. Soc., 1994).

  25. R. L. Dobrushin, Perturbation methods of the theory of Gibbsian fields, in Lectures on Probability Theory and Statistics (Saint-Flour, 1994), pp. 1-66, Lecture Notes in Math., Vol. 1648 (Springer, Berlin, 1996).

    Google Scholar 

  26. C. Gruber and H. Kunz, General properties of polymer systems,D. Ioffe, Self-avoiding polygons: Sharp asymptotics of canonical partition functions under the fixed area constraint, preprint (2001).

  27. O. Hryniv and D. Ioffe, Self-avoiding polygons: Sharp asymptotics of canonical partition functions under the fixed area constraint, preprint (2001).

  28. M. Pinsky, A note on the Erdös-Feller-Pollard theorem, Amer. Math. Monthly 83:729-731 (1976).

    Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Rights and permissions

Reprints and permissions

About this article

Cite this article

Hryniv, O., Kotecký, R. Surface Tension and the Ornstein–Zernike Behaviour for the 2D Blume–Capel Model. Journal of Statistical Physics 106, 431–476 (2002). https://doi.org/10.1023/A:1013797920029

Download citation

  • Issue Date:

  • DOI: https://doi.org/10.1023/A:1013797920029

Navigation