Communications in Mathematical Physics

, Volume 73, Issue 1, pp 83–94 | Cite as

Translation invariance and instability of phase coexistence in the two dimensional Ising system

  • Michael Aizenman


It is shown that any Gibbs state of the two dimensional ferromagnetic Ising system is of the form λμ++(1−λ)μ, with some λ∈ [0, 1]. This excludes the possibility of a locally stable phase coexistence and of translation symmetry breaking, which are known to occur in higher dimensions. Use is made in the proof of the stochastic aspects of the geometry of the interface lines.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Dobrushin, R.L.: Theor. Prob. Appl.17, 582 (1972)Google Scholar
  2. 2.
    Dobrushin, R.L.: Theor. Prob. Appl.13, 197 (1968)Google Scholar
  3. 2a.
    Lanford, O.E. III, Ruelle, D.: Commun. Math. Phys.13, 194 (1969)Google Scholar
  4. 3.
    Fortuin, C.M., Kasteleyn, P.W., Ginibre, J.: Commun. Math. Phys.22, 89 (1971)Google Scholar
  5. 4.
    Holley, R.: Commun. Math. Phys.36, 227 (1974)Google Scholar
  6. 5.
    van Beijeren, H.: Commun. Math. Phys.40, 1 (1975)Google Scholar
  7. 6.
    Lebowitz, J.L., Monroe, J.L.: Commun. Math. Phys.28, 301 (1972)Google Scholar
  8. 7.
    Bricmont, J., Lebowitz, J.L., Pfister, C.E., Olivieri, E.: Commun. Math. Phys.66, 1 (1979)Google Scholar
  9. 8.
    Gallavotti, G.: Commun. Math. Phys.27, 103 (1972)Google Scholar
  10. 9.
    Abraham, D.B., Reed, P.: Phys. Rev. Lett.33, 377 (1974)Google Scholar
  11. 10.
    Abraham, D.B., Reed, P.: Commun. Math. Phys.49, 35 (1976)Google Scholar
  12. 11.
    Messager, A., Miracle-Sole, S.: J. Stat. Phys.17, 245 (1977)Google Scholar
  13. 12.
    Coniglio, A., Nappi, C., Peruggi, F., Russo, L.: Commun. Math. Phys.51, 315 (1976)Google Scholar
  14. 13.
    Russo, L.: Commun. Math. Phys.67, 251 (1979)Google Scholar
  15. 14.
    Aizenman, M.: Phys. Rev. Lett.43, 407 (1979)Google Scholar
  16. 15.
    Merlini, D.: PreprintGoogle Scholar
  17. 16.
    Higuchi, Y.: In: Proc. Colloquium on Random Fields (Esztergom, June 1979). To appearGoogle Scholar

Copyright information

© Springer-Verlag 1980

Authors and Affiliations

  • Michael Aizenman
    • 1
  1. 1.Department of PhysicsPrinceton UniversityPrincetonUSA

Personalised recommendations