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Journal of Statistical Physics

, Volume 88, Issue 1–2, pp 177–200 | Cite as

Crumpling self-avoiding surfaces

  • E. J. Janse van Rensburg
Articles

Abstract

Self-avoiding plaquette surfaces with a folding or bending fugacity are believed to undergo a “crumpling” transition from a flaccid phase with branched polymer characteristics (corresponding to surfaces with a high degree of folding), to a “smooth” phase (corresponding to surfaces faith a low degree of folding). I develop rigorous techniques in order to bound the free energy of this model. In particular, the limiting free energy is proven to be positive for all positive values of the folding fugacity. In addition, the existence of a nonanalyticity in the limiting free energy of a (nontrivial) subclass of surfaces is proven. This implies the existence of a phase transition in this model, which I conjecture to be from a “flaccid” to a “smooth” phase.

Key Words

Plaquette surfaces fugacity flaccid phase smooth phase limiting free energy phase transition 

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References

  1. 1.
    G. Whittington,J. Math. Chem. 14:103 (1993).MathSciNetCrossRefGoogle Scholar
  2. 2.
    E. Orlandini, A. L. Stella, T. L. Einstein, M. C. Tesi, I. Beichl and F. Sullivan,Phys. Rev. E 53:5800 (1996).CrossRefADSGoogle Scholar
  3. 3.
    R. Lipowsky,Nature 349:475 (1991).CrossRefADSGoogle Scholar
  4. 4.
    E. Orlandini and M. C. Tesi,Physica A 185:160 (1992).CrossRefADSGoogle Scholar
  5. 5.
    A. Baumgartner,Physica A 192:550 (1993).CrossRefADSGoogle Scholar
  6. 6.
    E. Orlandini, A. L. Stella, M. C. Tesi and F. Sullivan,Phys. Rev. E 48:R4203 (1993).CrossRefADSGoogle Scholar
  7. 7.
    A. L. Stella, E. Orlandini, I. Beichl, F. Sullivan, M. C. Tesi and T. L. Einstein,Phys. Rev. Lett. 64:3650 (1992).CrossRefADSGoogle Scholar
  8. 8.
    U. Glaus,Phys. Rev. Lett. 56:1996 (1986).CrossRefADSGoogle Scholar
  9. 9.
    U. Glaus U.,J. Stat. Phys. 50:1141 (1988).CrossRefADSGoogle Scholar
  10. 10.
    U. Glaus and T. L. Einstein,J. Phys. A: Math. Gen. 20:L105 (1987).CrossRefADSGoogle Scholar
  11. 11.
    J. O'Connell, F. Sullivan, D. Libes, E. Orlandini, M. C. Tesi, A. L. Stella and T. L. Einstein,J. Phys. A: Math. Gen. 24:4619 (1991).CrossRefADSGoogle Scholar
  12. 12.
    A. Baumgartner and A. Romero,Physica A 187:243 (1992).CrossRefADSGoogle Scholar
  13. 13.
    M. E. Fisher, A. J. Guttmann and S. G. Whittington,J. Phys. A: Math. Gen. 24:3095 (1991).MathSciNetCrossRefADSGoogle Scholar
  14. 14.
    M. E. Fisher,Physics 3:255 (1967).Google Scholar
  15. 15.
    A. F. Andreev,Sov. Phys.-JETP 18:1415 (1964).Google Scholar
  16. 16.
    S. N. Isakov,Commun. Math. Phys. 95:427 (1984).MathSciNetCrossRefADSGoogle Scholar
  17. 17.
    A. L. Stella,Turkish J. Phys. 18:244 (1994).Google Scholar
  18. 18.
    E. J. Janse van Rensburg and S. G. Whittington,J. Phys. A: Math. Gen. 22:4939 (1989).zbMATHMathSciNetCrossRefADSGoogle Scholar
  19. 19.
    E. J. Janse van Rensburg and S. G. Whittington,J. Phys. A: Math. Gen. 23:2495 (1990).zbMATHMathSciNetCrossRefADSGoogle Scholar
  20. 20.
    N. Madras, C. E. Soteros, S. G. Whittington, J. L. Martin, M. F. Sykes, S. Flesia and D. S. Gaunt,J. Phys. A: Math. Gen. 23:5327 (1990).zbMATHMathSciNetCrossRefADSGoogle Scholar
  21. 21.
    N. Madras, C. E. Soteros and S. G. Whittington,J. Phys. A: Math. Gen. 21:4613 (1988).MathSciNetCrossRefADSGoogle Scholar
  22. 22.
    E. J. Janse van Rensburg,J. Phys. A: Math. Gen. 23:5879 (1990).zbMATHMathSciNetCrossRefGoogle Scholar
  23. 23.
    E. J. Janse van Rensburg,J. Phys. A: Math. Gen. 25:3529 (1992).zbMATHMathSciNetCrossRefADSGoogle Scholar
  24. 24.
    E. J. Janse van Rensburg,J. Knot Theory Ram. 3:365 (1994).zbMATHMathSciNetCrossRefGoogle Scholar
  25. 25.
    B. Durhuus, J. Fröhlich and T. Jonsson,Nucl. Phys. B225[FS9] 185 (1983).CrossRefADSGoogle Scholar
  26. 26.
    J. Ginibre, A. Grossmann and D. Ruelle,Commun. Math. Phys. 3:18 (1966).CrossRefGoogle Scholar
  27. 27.
    E. Hille,Functional Analysis and Semi-Groups (AMS Colloq. Publ.)31 (New York: AMS) (1948).zbMATHGoogle Scholar
  28. 28.
    J. B. Wilker and S. G. Whittington,J. Phys. A: Math. Gen. 12:L245 (1979).MathSciNetCrossRefADSGoogle Scholar
  29. 29.
    D. Klarner,Can. J. Math. 19:851 (1967).zbMATHMathSciNetGoogle Scholar
  30. 30.
    R. Buck,Advanced Calculus 2nd ed. McGraw-Hill, New York 1965 (1965).zbMATHGoogle Scholar
  31. 31.
    M. E. Fisher,J. Math. Phys. 6:1643. Colloq. Publ.31 (New York: AMS) (1965).CrossRefGoogle Scholar

Copyright information

© Plenum Publishing Corporation 1997

Authors and Affiliations

  • E. J. Janse van Rensburg
    • 1
  1. 1.Department of Mathematics and StatisticsYork UniversityNorth YorkCanada

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