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.
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References
G. Whittington,J. Math. Chem. 14:103 (1993).
E. Orlandini, A. L. Stella, T. L. Einstein, M. C. Tesi, I. Beichl and F. Sullivan,Phys. Rev. E 53:5800 (1996).
R. Lipowsky,Nature 349:475 (1991).
E. Orlandini and M. C. Tesi,Physica A 185:160 (1992).
A. Baumgartner,Physica A 192:550 (1993).
E. Orlandini, A. L. Stella, M. C. Tesi and F. Sullivan,Phys. Rev. E 48:R4203 (1993).
A. L. Stella, E. Orlandini, I. Beichl, F. Sullivan, M. C. Tesi and T. L. Einstein,Phys. Rev. Lett. 64:3650 (1992).
U. Glaus,Phys. Rev. Lett. 56:1996 (1986).
U. Glaus U.,J. Stat. Phys. 50:1141 (1988).
U. Glaus and T. L. Einstein,J. Phys. A: Math. Gen. 20:L105 (1987).
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).
A. Baumgartner and A. Romero,Physica A 187:243 (1992).
M. E. Fisher, A. J. Guttmann and S. G. Whittington,J. Phys. A: Math. Gen. 24:3095 (1991).
M. E. Fisher,Physics 3:255 (1967).
A. F. Andreev,Sov. Phys.-JETP 18:1415 (1964).
S. N. Isakov,Commun. Math. Phys. 95:427 (1984).
A. L. Stella,Turkish J. Phys. 18:244 (1994).
E. J. Janse van Rensburg and S. G. Whittington,J. Phys. A: Math. Gen. 22:4939 (1989).
E. J. Janse van Rensburg and S. G. Whittington,J. Phys. A: Math. Gen. 23:2495 (1990).
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).
N. Madras, C. E. Soteros and S. G. Whittington,J. Phys. A: Math. Gen. 21:4613 (1988).
E. J. Janse van Rensburg,J. Phys. A: Math. Gen. 23:5879 (1990).
E. J. Janse van Rensburg,J. Phys. A: Math. Gen. 25:3529 (1992).
E. J. Janse van Rensburg,J. Knot Theory Ram. 3:365 (1994).
B. Durhuus, J. Fröhlich and T. Jonsson,Nucl. Phys. B225[FS9] 185 (1983).
J. Ginibre, A. Grossmann and D. Ruelle,Commun. Math. Phys. 3:18 (1966).
E. Hille,Functional Analysis and Semi-Groups (AMS Colloq. Publ.)31 (New York: AMS) (1948).
J. B. Wilker and S. G. Whittington,J. Phys. A: Math. Gen. 12:L245 (1979).
D. Klarner,Can. J. Math. 19:851 (1967).
R. Buck,Advanced Calculus 2nd ed. McGraw-Hill, New York 1965 (1965).
M. E. Fisher,J. Math. Phys. 6:1643. Colloq. Publ.31 (New York: AMS) (1965).
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Janse van Rensburg, E.J. Crumpling self-avoiding surfaces. J Stat Phys 88, 177–200 (1997). https://doi.org/10.1007/BF02508469
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DOI: https://doi.org/10.1007/BF02508469