Communications in Mathematical Physics

, Volume 242, Issue 1–2, pp 137–183 | Cite as

Critical Region for Droplet Formation in the Two-Dimensional Ising Model

  • Marek Biskup
  • Lincoln Chayes
  • Roman Kotecký


We study the formation/dissolution of equilibrium droplets in finite systems at parameters corresponding to phase coexistence. Specifically, we consider the 2D Ising model in volumes of size L 2 , inverse temperature β>β c and overall magnetization conditioned to take the value m L 2 −2m v L , where β c −1 is the critical temperature, m =m (β) is the spontaneous magnetization and v L is a sequence of positive numbers. We find that the critical scaling for droplet formation/dissolution is when v L 3/2 L −2 tends to a definite limit. Specifically, we identify a dimensionless parameter Δ, proportional to this limit, a non-trivial critical value Δ c and a function λΔ such that the following holds: For Δ<Δ c , there are no droplets beyond log L scale, while for Δ>Δ c , there is a single, Wulff-shaped droplet containing a fraction λΔ≥λ c =2/3 of the magnetization deficit and there are no other droplets beyond the scale of log L. Moreover, λΔ and Δ are related via a universal equation that apparently is independent of the details of the system.


Critical Temperature Dimensionless Parameter Critical Region Ising Model Inverse Temperature 
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  1. 1.
    Abraham, D.B., Martin-Löf, A.: The transfer matrix for a pure phase in the two-dimensional Ising model. Commun. Math. Phys. 31, 245–268 (1973)Google Scholar
  2. 2.
    Aizenman, M., Chayes, J.T., Chayes, L., Newman, C.M.: Discontinuity of the magnetization in one-dimensional 1/| x-y|2 Ising and Potts models. J. Stat. Phys. 50, 1–40 (1988)MathSciNetzbMATHGoogle Scholar
  3. 3.
    Alexander, K.: Cube-root boundary fluctuations for droplets in random cluster models. Commun. Math. Phys. 224, 733–781 (2001)CrossRefMathSciNetzbMATHGoogle Scholar
  4. 4.
    Alexander, K., Chayes, J.T., Chayes, L.: The Wulff construction and asymptotics of the finite cluster distribution for two-dimensional Bernoulli percolation. Commun. Math. Phys. 131, 1–51 (1990)MathSciNetzbMATHGoogle Scholar
  5. 5.
    Ben Arous, G., Deuschel, J.-D.: The construction of the d+1-dimensional Gaussian droplet. Commun. Math. Phys. 179, 467–488 (1996)zbMATHGoogle Scholar
  6. 6.
    Bennetin, G., Gallavotti, G., Jona-Lasinio, G., Stella, A.: On the Onsager-Yang value of the spontaneous magnetization. Commun. Math. Phys. 30, 45–54 (1973)Google Scholar
  7. 7.
    Binder, K.: Theory of evaporation/condensation transition of equilibrium droplets in finite volumes. Physica A 319, 99–114 (2003)CrossRefGoogle Scholar
  8. 8.
    Binder, K.: Reply to ‘Comment on ‘‘Theory of the evaporation/condensation transition of equilibrium droplets in finite volumes’’’. Physica A 327, 589–592 (2003)CrossRefGoogle Scholar
  9. 9.
    Binder, K., Kalos, M.H.: Critical clusters in a supersaturated vapor: Theory and Monte Carlo simulation. J. Statist. Phys. 22, 363–396 (1980)Google Scholar
  10. 10.
    Biskup, M., Chayes, L., Kotecký, R.: On the formation/dissolution of equilibrium droplets. Europhys. Lett. 60(1), 21–27 (2002)Google Scholar
  11. 11.
    Biskup, M., Chayes, L., Kotecký, R.: Comment on ‘‘Theory of the evaporation/condensation tran- sition of equilibrium droplets in finite volumes’’. Physica A 327, 583–588 (2003)CrossRefGoogle Scholar
  12. 12.
    Biskup, M., Borgs, C., Chayes, J.T., Kotecký, R.: Gibbs states of graphical representations of the Potts model with external fields. J. Math. Phys. 41, 1170–1210 (2000)CrossRefMathSciNetzbMATHGoogle Scholar
  13. 13.
    Bodineau, T.: The Wulff construction in three and more dimensions. Commun. Math. Phys. 207, 197–229 (1999)CrossRefMathSciNetzbMATHGoogle Scholar
  14. 14.
    Bodineau, T., Ioffe, D., Velenik, Y.: Rigorous probabilistic analysis of equilibrium crystal shapes. J. Math. Phys. 41, 1033–1098 (2000)CrossRefMathSciNetzbMATHGoogle Scholar
  15. 15.
    Bolthausen E., Ioffe, D.: Harmonic crystal on the wall: a microscopic approach. Commun. Math. Phys. 187, 523–566 (1997)CrossRefMathSciNetzbMATHGoogle Scholar
  16. 16.
    Borgs, C., Kotecký, R.: Surface-induced finite-size effects for first-order phase transitions. J. Stat. Phys. 79, 43–115 (1995)MathSciNetzbMATHGoogle Scholar
  17. 17.
    Bricmont, J., Lebowitz, J.L., Pfister, C.E.: On the local structure of the phase separation line in the two-dimensional Ising system. J. Statist. Phys. 26(2), 313–332 (1981)Google Scholar
  18. 18.
    Campanino, M., Chayes, J.T., Chayes, L.: Gaussian fluctuations of connectivities in the subcritical regime of percolation. Probab. Theory Rel. Fields 88, 269–341 (1991)MathSciNetzbMATHGoogle Scholar
  19. 19.
    Campanino, M., Ioffe, D.: Ornstein-Zernike theory for the Bernoulli bond percolation on Z d. Ann. Probab. 30(2), 652–682 (2002)zbMATHGoogle Scholar
  20. 20.
    Campanino, M., Ioffe, D., Velenik, Y.: Ornstein-Zernike theory for the finite-range Ising models above T c. Probab. Theory Rel. Fields 125(3), 305–349 (2003)zbMATHGoogle Scholar
  21. 21.
    Cerf, R.: Large deviations for three dimensional supercritical percolation. Astérisque 267, vi+177 (2000)Google Scholar
  22. 22.
    Cerf, R., Pisztora, A.: On the Wulff crystal in the Ising model. Ann. Probab. 28, 947–1017 (2000)CrossRefMathSciNetzbMATHGoogle Scholar
  23. 23.
    Chayes, J.T., Chayes, L., Fisher, D.S., Spencer, T.: Correlation length bounds for disordered Ising ferromagnets. Commun. Math. Phys. 120, 501–523 (1989)MathSciNetzbMATHGoogle Scholar
  24. 24.
    Chayes, J.T., Chayes, L., Schonmann, R.H.: Exponential decay of connectivities in the two-dimensional Ising model. J. Statist. Phys. 49, 433–445 (1987)MathSciNetzbMATHGoogle Scholar
  25. 25.
    Curie, P.: Sur la formation des cristaux et sur les constantes capillaires de leurs différentes faces. Bull. Soc. Fr. Mineral. 8, 145 (1885); Reprinted in Œuvres de Pierre Curie, Paris: Gauthier-Villars, 1908, pp. 153–157Google Scholar
  26. 26.
    Dobrushin, R.L., Hryniv, O.: Fluctuation of the phase boundary in the 2D Ising ferromagnet. Commun. Math. Phys. 189, 395–445 (1997)CrossRefMathSciNetzbMATHGoogle Scholar
  27. 27.
    Dobrushin, R.L., Kotecký, R., Shlosman, S.B.: Wulff construction. A global shape from local interaction. Providence, RI: Am. Math. Soc., 1992Google Scholar
  28. 28.
    Dobrushin, R.L., Shlosman, S.B.: Large and moderate deviations in the Ising model. In: Probability contributions to statistical mechanics, Adv. Soviet Math., Vol. 20, Providence, RI: Amer. Math. Soc., 1994, pp. 91–219Google Scholar
  29. 29.
    Dunlop, F., Magnen, J., Rivasseau, V., Roche, Ph.: Pinning of an interface by a weak potential. J. Statist. Phys. 66, 71–98 (1992)MathSciNetzbMATHGoogle Scholar
  30. 30.
    Edwards, R.G., Sokal, A.D.: Generalization of the Fortuin-Kasteleyn-Swendsen-Wang representation and Monte Carlo algorithm. Phys. Rev. D 38, 2009–2012 (1988)CrossRefMathSciNetGoogle Scholar
  31. 31.
    Georgii, H.-O.: Gibbs Measures and Phase Transitions. de Gruyter Studies in Mathematics, Vol. 9, Berlin: Walter de Gruyter & Co., 1988Google Scholar
  32. 32.
    Georgii, H.-O., Häggström, O., Maes, C.: The random geometry of equilibrium phases. In: C. Domb and J.L. Lebowitz (eds), Phase Transitions and Critical Phenomena, Vol. 18, New York: Academic Press, 1999, pp. 1–142Google Scholar
  33. 33.
    Gibbs, J.W.: On the equilibrium of heterogeneous substances. (1876). In: Collected Works, Vol. 1, London: Longmans, Green and Co., 1928Google Scholar
  34. 34.
    Griffiths, R.B., Hurst, C.A., Sherman, S.: Concavity of magnetization of an Ising ferromagnet in a positive external field. J. Math. Phys. 11, 790–795 (1970)Google Scholar
  35. 35.
    Grimmett, G.R.: The stochastic random cluster process and the uniqueness of random cluster measures. Ann. Probab. 23, 1461–1510 (1995)MathSciNetzbMATHGoogle Scholar
  36. 36.
    Gross, D.H.E.: Microcanonical Thermodynamics: Phase Transitions in ‘‘Small’’ Systems. Lecture Notes in Physics, Vol. 66, Singapore: World Scientific, 2001Google Scholar
  37. 37.
    Hryniv, O., Kotecký, R.: Surface tension and the Ornstein-Zernike behaviour for the 2D Blume-Capel model. J. Stat. Phys. 106(3-4), 431–476 (2002)Google Scholar
  38. 38.
    Ioffe, D.: Large deviations for the 2D Ising model: a lower bound without cluster expansions. J. Statist. Phys. 74, 411–432 (1994)MathSciNetzbMATHGoogle Scholar
  39. 39.
    Ioffe, D.: Exact large deviation bounds up to T c for the Ising model in two dimensions. Probab. Theory Rel. Fields 102, 313–330 (1995)MathSciNetzbMATHGoogle Scholar
  40. 40.
    Ioffe, D., Schonmann, R.H.: Dobrushin-Kotecký-Shlosman theorem up to the critical temperature. Commun. Math. Phys. 199, 117–167 (1998)CrossRefMathSciNetzbMATHGoogle Scholar
  41. 41.
    Kaufman, B., Onsager, L.: Crystal statistics. III. Short range order in a binary Ising lattice. Phys. Rev. 76, 1244–1252 (1949)CrossRefzbMATHGoogle Scholar
  42. 42.
    Krishnamachari, B., McLean, J., Cooper, B., Sethna, J.: Gibbs-Thomson formula for small island sizes: Corrections for high vapor densities. Phys. Rev. B 54, 8899–8907 (1996)CrossRefGoogle Scholar
  43. 43.
    Lee, J., Kosterlitz, J.M.: Finite-size scaling and Monte Carlo simulations of first-order phase transitions. Phys. Rev. B 43, 3265–3277 (1990)CrossRefGoogle Scholar
  44. 44.
    Machta, J., Choi, Y.S., Lucke, A., Schweizer, T., Chayes, L.M.: Invaded cluster algorithm for Potts models. Phys. Rev. E 54, 1332–1345 (1996)CrossRefGoogle Scholar
  45. 45.
    Müller, T., Selke, W.: Stability and diffusion of surface clusters. Eur. Phys. J. B 10, 549–553 (1999)CrossRefGoogle Scholar
  46. 46.
    Neuhaus, T., Hager, J.S.: 2d crystal shapes, droplet condensation and supercritical slowing down in simulations of first order phase transitions. J. Statist. Phys. 113, 47–83 (2003)CrossRefGoogle Scholar
  47. 47.
    Onsager, L.: Crystal statistics. I. A two-dimensional model with an order-disorder transition. Phys. Rev. 65, 117–149 (1944)CrossRefzbMATHGoogle Scholar
  48. 48.
    Pfister, C.-E.: Large deviations and phase separation in the two-dimensional Ising model. Helv. Phys. Acta. 64, 953–1054 (1991)MathSciNetGoogle Scholar
  49. 49.
    Pfister, C.-E., Velenik, Y.: Large deviations and continuum limit in the 2D Ising model. Probab. Theory Rel. Fields 109, 435–506 (1997)CrossRefMathSciNetzbMATHGoogle Scholar
  50. 50.
    Pfister, C.-E., Velenik, Y.: Interface, surface tension and reentrant pinning transition in 2D Ising model. Commun. Math. Phys. 204, 269–312 (1999)CrossRefMathSciNetzbMATHGoogle Scholar
  51. 51.
    Pleimling, M., Hüller, A.: Crossing the coexistence line at constant magnetization. J. Statist. Phys. 104, 971–989 (2001)CrossRefzbMATHGoogle Scholar
  52. 52.
    Pleimling, M., Selke, W.: Droplets in the coexistence region of the two-dimensional Ising model. J. Phys. A: Math. Gen. 33, L199–L202 (2000)Google Scholar
  53. 53.
    Schonmann, R.H., Shlosman, S.B.: Wulff droplets and the metastable relaxation of kinetic Ising models. Commun. Math. Phys. 194, 389–462 (1998)CrossRefMathSciNetzbMATHGoogle Scholar
  54. 54.
    Simon, B.: The Statistical Mechanics of Lattice Gases. Vol. I., Princeton Series in Physics, Princeton, NJ: Princeton University Press, 1993.Google Scholar
  55. 55.
    Wulff, G.: Zur Frage des Geschwindigkeit des Wachsturms und der Auflösung der Krystallflachen. Z. Krystallog. Mineral. 34, 449–530 (1901)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2003

Authors and Affiliations

  • Marek Biskup
    • 1
  • Lincoln Chayes
    • 1
  • Roman Kotecký
    • 2
  1. 1.Department of MathematicsUCLALos AngelesUSA
  2. 2.Center for Theoretical StudyCharles UniversityPragueCzech Republic

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