Survey of Self-Avoiding Random Surfaces on Cubic Lattices: Issues, Controversies, and Results*

  • T. L. Einsteint
  • A. L. Stella
Part of the The IMA Volumes in Mathematics and its Applications book series (IMA, volume 103)


The aim of this short review is to present an overview of problems involving random surfaces on a cubic lattice: why one studies them, what questions one tries to answer, what assumptions are made in the models, what one computes to understand these models, and some things that have been learned. The emphasis will be on what one can calculate and why the problem is computationally intensive but tractable. There has also been considerable progress in rigorous treatments of this subject; this train of work will be discussed in the following paper. Since there have been some rather extensive reviews [1][2][3][4] on the subject, there is no need to be encyclopedic. Liberal use is made of these sources as well as unpublished lecture notes by A. Maritan [5] and E. Orlandini [6].


Domain Wall Osmotic Pressure Critical Exponent Universality Class Lattice Gauge Theory 
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.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. [1]
    J. Fröhlich, “The Statistical Mechanics of Surfaces,” in: Applications of Field Theory to Statistical Mechanics (Lectures Notes in Physics 216), ed. by L. garrido (Springer, Berlin, 1985), 31–57.CrossRefGoogle Scholar
  2. [2]
    U. Glaus, “Monte Carlo Study of Self Avoiding Surfaces,” J. Stat. Phys. 50, 1141–1166 (1988).CrossRefGoogle Scholar
  3. [3]
    A.L. StellaSelf Avoiding Surfaces and Vesicles on a Lattice,” in: Complex Systems: Fractals, Spin Glasses and Neural Networks ed. by G. Parisi, L. Pietronero, and M. VirasoroPhysica A 185 211–221 (1992).Google Scholar
  4. [4]
    A.L. Stella “Statistical Mechanics of Random Surfaces, Vesicles and Polymers,” Turkish J. Phys. 18, 244–260 (1994).Google Scholar
  5. [5]
    A. Maritan, unpublished lecture notes.Google Scholar
  6. [6]
    E. Orlandini, unpublished lecture notes.Google Scholar
  7. [7]
    A. Maritan and A.L. Stella, “Some Exact Results for Self-Avoiding Random Surfaces,” Nucl. Phys. 280, 561–575 (1987).MathSciNetCrossRefGoogle Scholar
  8. [8]
    P.G. de Gennes, Scaling Concepts in Polymer Physics (Cornell University Press, Ithaca, 1979).Google Scholar
  9. [9]
    P.M. Chaikin and T.C. Lubensky, Principles of Condensed Matter Physics (Cambridge University Press, Cambridge, 1995).Google Scholar
  10. [10]
    S.G. Whittington, “Statistical Mechanics of Three Dimensional Vesicles,” J. Math. Chem. 14, 103–110 (1993).MathSciNetCrossRefGoogle Scholar
  11. [11]
    B. Durhuus, J. Fröhlich and T. Jonsson, “Self-Avoiding and Planar Random Surfaces on the Lattice,” Nucl. Phys. B 225 [FS9], 185–203 (1983).CrossRefGoogle Scholar
  12. [12]
    J.M. Hammersley, “The Number of Polygons on a Lattice,” Proc. Camb. Phil. Soc. 57, 516–523 (1961).MathSciNetzbMATHCrossRefGoogle Scholar
  13. [13]
    A. Baumgartner, “Inflated Vesicles: A Lattice Model,” Physica A 190, 63–74 (1992).CrossRefGoogle Scholar
  14. [14]
    A. Baumgartner and A. Romero, “Microcanonical Simulation of Self-Avoiding Surfaces,” Physica A 187, 243–248 (1992).CrossRefGoogle Scholar
  15. [15]
    A. Baumgartner, “Phase Transition of Semiflexible Lattice Vesicles,” Physica A 192, 550–561 (1993).CrossRefGoogle Scholar
  16. [16]
    J. O’connell, F. Sullivan, D. Liges, E. Orlandini, M.C. Tesl, A.L. Stella and T.L. Einstein, “Self-Avoiding Random Surfaces: Monte Carlo Study Using Oct-Tree Data-Structures,” J. Phys. A 24, 4619–4635 (1991).CrossRefGoogle Scholar
  17. [17]
    T. Sterling and J. Greensite, “Entropy of Self-Avoiding Surfaces on the Lattice,” Phys. Lett. B 121, 345–348 (1983).Google Scholar
  18. [18]
    U. Glaus and T.L. Einstein, “On the Universality Class of Planar Self-Avoiding Surfaces with Fixed Boundary,” J. Phys. A 20, L105–L111 (1987).CrossRefGoogle Scholar
  19. [19]
    E. Orlandini, Ph.D. Thesis, U. of Bologna, 1993 (unpublished).Google Scholar
  20. [20]
    B. Durhuus, J. Fröhlich and T. Jonsson, “Critical Behaviour in a Model of Planar Random Surfaces,” Nucl. Phys. B 240 [FS12], 453–480 (1984).CrossRefGoogle Scholar
  21. [21]
    J.M. Drouffe, G. Parisi and N. Sourlas, “Strong Coupling Phase in Lattice Gauge Theories at Large Dimension,” Nucl. Phys. B 161, 397–416 (1980).CrossRefGoogle Scholar
  22. [22]
    B. Baumann and B. Berg, “Non-Trivial Lattice Random Surfaces,” Phys. Lett. 164B, 131–135 (1985).Google Scholar
  23. [23]
    A. Maritan and A.L. Stella, “Scaling Behavior of Self-Avoiding Random Surfaces,” Phys. Rev. Lett. 53, 123–126 (1984).CrossRefGoogle Scholar
  24. [24]
    G. Parisi and N. Sourlas, “Critical Behavior of Branched Polymers and the Lee-Yang Edge Singularity,” Phys. Rev. Lett. 46, 871–874 (1981).MathSciNetCrossRefGoogle Scholar
  25. [25]
    S. Redner, “Enumeration Study of Self-Avoiding Random Surfaces,” J. Phys. A 18, L723 L733(1985); 19, 3199 (E) (1986).MathSciNetGoogle Scholar
  26. [26a]
    M.E. Cates, “The Fractal Dimension and Connectivity of Random Surfaces,” Phys. Lett. 161B, 363–367 (1985);MathSciNetGoogle Scholar
  27. [26b]
    H. Tasaki and T. Hara, “Collapse of Random Surfaces in the Connected Plaquettes Model,” Phys. Lett. 112A 115–118 (1985).MathSciNetGoogle Scholar
  28. [27]
    M.E. Fisher, A.J. Guttmann and S.G. Whittington, “Two-Dimensional Lattice Vesicles and Polygons,” J. Phys. A 24, 3095–3106 (1991).MathSciNetCrossRefGoogle Scholar
  29. [28]
    A.L. Stella, E. Orlandini, I. Beichl, F. Sullivan, M.C. Tesi and T.L. Einstein, “Self-Avoiding Surfaces, Topology, and Lattice Animals,” Phys. Rev. Lett. 69, 3650–3653 (1992).CrossRefGoogle Scholar
  30. [29]
    E. Orlandini, A.L. Stella, T.L. Einstein, M.C. Tesi, I. Beichl and F. Sullivan, “Bending-Rigidity-Driven Transitions and Crumpling-Point Scaling of Lattice Vesicles,” Phys. Rev. E 53, 5800–5807 (1996).CrossRefGoogle Scholar
  31. [30]
    E.J. Janse Van Rensburg, “Crumpling Self-Avoiding Surfaces,” J. Stat. Phys., Vol. 88 - No. 1/2, July 1997.Google Scholar
  32. [31a]
    J. Banavar, A. Maritan and A. Stella, “Geometry, Topology, and Universality of Random Surfaces,” Science 252, 825--827 (1991);MathSciNetCrossRefGoogle Scholar
  33. [31b]
    “Critical Behavior of Two Dimensional Vesicles in the Deflated Regime,” Phys. Rev. A 43, R5752 R5754 (1991) for the 2D case.Google Scholar
  34. [32]
    C. Soteros and S. Whittington, “Critical Exponents for Lattice Animals with Fixed Cyclomatic Index,” J. Phys. A: Math. Gen. 21, 2187–2193 (1988).MathSciNetCrossRefGoogle Scholar
  35. [33]
    E. Orlandini, A.L. Stella, M.C. Tesi and F. Sullivan, “Vesicle Adsorption on a Plane: Scaling Regimes and Crossover Phenomena,” Phys. Rev. E 48, R4203–R4206 (1993).CrossRefGoogle Scholar
  36. [34]
    K. Binder, “Critical Behaviour at Surfaces,” in: Phase Transitions and Critical Phenomena, vol. 8, ed. by C. Domb and J.L. Lebowitz (Academic, New York, 1983), chap. 1, 1–144.Google Scholar

Copyright information

© Springer-Verlag New York, Inc. 1998

Authors and Affiliations

  • T. L. Einsteint
    • 1
  • A. L. Stella
    • 2
  1. 1.Department of PhysicsUniversity of Maryland, College ParkUSA
  2. 2.INFM-Dipartimento di Fisica e Sezione INFN dell’ Università. di PadovaPadovaItaly

Personalised recommendations