Abstract
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.
Similar content being viewed by others
References
Abraham, D. B.: In Phase transitions and critical phenomena, Vol. 10. Domb, C., Lebowitz, J. L. (eds.). London: Academic Press 1986
Aizenman, M., Barsky, D. J.: Commun. Math. Phys.108, 489 (1987)
Abraham, D. B., Martin-Löf, A.: Commun. Math. Phys.49, 35 (1976)
Abraham, D. B., Newman, C. M.: Phys. Rev. Lett.61, 1969 (1988)
Abraham, D. B., Reed, P.: J. Phys. A10, L121 (1977)
van Beijeren, H.: Commun. Math. Phys.40, 1 (1975)
van Beijeren, H.: Phys. Rev. Lett.38, 993 (1977)
Burton, W. K., Cabrera, N., Frank, F. C.: Philos. Trans. Roy. Soc. (London) A243, 299 (1951)
Bricmont, J., El Mellouki, A., Fröhlich, J.: J. Stat. Phys.42, 743 (1986)
Benettin, G., Gallavotti, G., Jona-Lasinio, G., Stella, A.: Commun. Math. Phys.30, 45 (1973)
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: Springer
Coniglio, A., Nappi, C. R., Peruggi, F., Russo, L.: Commun. Math. Phys.51, 315 (1976)
Dietrich, S.: In: Phase transitions and critical phenomena, vol. 12. Domb, C., Lebowitz, J. L. (eds.) London: Academic Press 1988
Dobrushin, R. L.: Th. Prob. Appl.17, 582 (1972);18, 253 (1973)
Esary, J. D., Proschan, F., Walkup, D. W.: Ann. Math. Stat.38, 1466 (1967)
Forrester, P. J.: J. Phys. A19, L143 (1986)
Fisher, D. S., Fisher, M. E.: Phys. Rev.25, 3192 (1982)
Fortuin, C. M., Kasteleyn, P. W., Ginibre, J.: Commun. Math. Phys.22, 89 (1971)
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 1983
Gandolfi, A., Keane, M., Russo, L.: Ann. Probab.16, 1147 (1988)
Hammersley, J. M.: Ann. Math. Stat.28, 790 (1957)
Jogdeo, K.: Ann. Stat.6, 232 (1978)
Kossel, W.: Nachr. Ges. Wiss. Göttingen, Mathemat./Physikal. Klasse S. 135 (1927)
Lieb, E. H.: Phys. Rev. Lett.18, 1046 (1967)
Lieb, E. H., Wu, F. Y.: In: Phase transitions and critical phenomena, vol. 1. Domb, C., Green, M. S. (eds.). London: Academic Press 1972
Menshikov, M. V.: Sov. Math. Dokl.33, 856 (1986)
Onsager, L.: Phys. Rev.65, 117 (1944)
Russo, L.: Commun. Math. Phys.67, 251 (1979)
Stransky, J. N.: Z. Phys. Chemie136, 259 (1928)
Toth, B.: Z. für Wahrscheinlichkeitstheorie69, 19 (1985)
Weeks, J. D.: In: Ordering in strongly fluctuating condensed matter systems. Riste, T. (ed.). New York: Plenum Press 1983
Yang, C. N.: Phys. Rev.85, 809 (1952)
Author information
Authors and Affiliations
Additional information
Communicated by A. Jaffe
Dedicated to Roland Dobrushin
Research supported in part by NSF Grant No. DMS-8514834 and AFOSR Contract No. F49620-86-C0130 under the U.R.I. program
Rights and permissions
About this article
Cite this article
Abraham, D.B., Newman, C.M. Surfaces and Peierls contours: 3-d wetting and 2-d Ising percolation. Commun.Math. Phys. 125, 181–200 (1989). https://doi.org/10.1007/BF01217776
Received:
Issue Date:
DOI: https://doi.org/10.1007/BF01217776