Communications in Mathematical Physics

, Volume 125, Issue 1, pp 181–200 | Cite as

Surfaces and Peierls contours: 3-d wetting and 2-d Ising percolation

  • D. B. Abraham
  • C. M. Newman


A natural model of a discrete random surface lying above a two-dimensional substrate is presented and analyzed. An identification of the “level curves” of the surface with the Peierls contours of Ising spin configurations leads to an exactly solvable free energy, with logarithmically divergent specific heat. The thermodynamic critical point is shown to be a wetting transition at which the surface height diverges. This is so even though the surface has no “downward fingers” and hence no “entropic repulsion” from the substrate.


Neural Network Free Energy Statistical Physic Complex System Nonlinear Dynamics 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


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  1. [A]
    Abraham, D. B.: In Phase transitions and critical phenomena, Vol. 10. Domb, C., Lebowitz, J. L. (eds.). London: Academic Press 1986Google Scholar
  2. [AB]
    Aizenman, M., Barsky, D. J.: Commun. Math. Phys.108, 489 (1987)Google Scholar
  3. [AM]
    Abraham, D. B., Martin-Löf, A.: Commun. Math. Phys.49, 35 (1976)Google Scholar
  4. [AN]
    Abraham, D. B., Newman, C. M.: Phys. Rev. Lett.61, 1969 (1988)Google Scholar
  5. [AR]
    Abraham, D. B., Reed, P.: J. Phys. A10, L121 (1977)Google Scholar
  6. [B1]
    van Beijeren, H.: Commun. Math. Phys.40, 1 (1975)Google Scholar
  7. [B2]
    van Beijeren, H.: Phys. Rev. Lett.38, 993 (1977)Google Scholar
  8. [BCF]
    Burton, W. K., Cabrera, N., Frank, F. C.: Philos. Trans. Roy. Soc. (London) A243, 299 (1951)Google Scholar
  9. [BEF]
    Bricmont, J., El Mellouki, A., Fröhlich, J.: J. Stat. Phys.42, 743 (1986)Google Scholar
  10. [BGJS]
    Benettin, G., Gallavotti, G., Jona-Lasinio, G., Stella, A.: Commun. Math. Phys.30, 45 (1973)Google Scholar
  11. [BN]
    van Beijeren, H., Nolden, I.: The roughening transition. In Structure and dynamics of interfaces II, vol. 43 of current topics in physics. Berlin, Heidelberg, New York: SpringerGoogle Scholar
  12. [CNPR]
    Coniglio, A., Nappi, C. R., Peruggi, F., Russo, L.: Commun. Math. Phys.51, 315 (1976)Google Scholar
  13. [D]
    Dietrich, S.: In: Phase transitions and critical phenomena, vol. 12. Domb, C., Lebowitz, J. L. (eds.) London: Academic Press 1988Google Scholar
  14. [Do]
    Dobrushin, R. L.: Th. Prob. Appl.17, 582 (1972);18, 253 (1973)Google Scholar
  15. [EPW]
    Esary, J. D., Proschan, F., Walkup, D. W.: Ann. Math. Stat.38, 1466 (1967)Google Scholar
  16. [F]
    Forrester, P. J.: J. Phys. A19, L143 (1986)Google Scholar
  17. [FF]
    Fisher, D. S., Fisher, M. E.: Phys. Rev.25, 3192 (1982)Google Scholar
  18. [FKG]
    Fortuin, C. M., Kasteleyn, P. W., Ginibre, J.: Commun. Math. Phys.22, 89 (1971)Google Scholar
  19. [FS]
    Fröhlich, J., Spencer, T.: Phys. Rev. Lett.46, 4006 (1981); Commun. Math. Phys.81, 527 (1981); In: Scaling and self-similarity in physics. Fröhlich, J. (ed.). Basel: Birkhäuser 1983Google Scholar
  20. [GKR]
    Gandolfi, A., Keane, M., Russo, L.: Ann. Probab.16, 1147 (1988)Google Scholar
  21. [H]
    Hammersley, J. M.: Ann. Math. Stat.28, 790 (1957)Google Scholar
  22. [J]
    Jogdeo, K.: Ann. Stat.6, 232 (1978)Google Scholar
  23. [K]
    Kossel, W.: Nachr. Ges. Wiss. Göttingen, Mathemat./Physikal. Klasse S. 135 (1927)Google Scholar
  24. [L]
    Lieb, E. H.: Phys. Rev. Lett.18, 1046 (1967)Google Scholar
  25. [LW]
    Lieb, E. H., Wu, F. Y.: In: Phase transitions and critical phenomena, vol. 1. Domb, C., Green, M. S. (eds.). London: Academic Press 1972Google Scholar
  26. [M]
    Menshikov, M. V.: Sov. Math. Dokl.33, 856 (1986)Google Scholar
  27. [O]
    Onsager, L.: Phys. Rev.65, 117 (1944)Google Scholar
  28. [R]
    Russo, L.: Commun. Math. Phys.67, 251 (1979)Google Scholar
  29. [S]
    Stransky, J. N.: Z. Phys. Chemie136, 259 (1928)Google Scholar
  30. [T]
    Toth, B.: Z. für Wahrscheinlichkeitstheorie69, 19 (1985)Google Scholar
  31. [W]
    Weeks, J. D.: In: Ordering in strongly fluctuating condensed matter systems. Riste, T. (ed.). New York: Plenum Press 1983Google Scholar
  32. [Y]
    Yang, C. N.: Phys. Rev.85, 809 (1952)Google Scholar

Copyright information

© Springer-Verlag 1989

Authors and Affiliations

  • D. B. Abraham
    • 1
  • C. M. Newman
    • 2
  1. 1.Department of Theoretical ChemistryOxfordEngland
  2. 2.Department of Mathematics and Ariz. Center for Math. SciencesUniversity of ArizonaTucsonUSA

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