Topology and Geometry in Polymer Science pp 159-173 | Cite as

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

## Abstract

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 Liberal use is made of these sources as well as unpublished lecture notes by A. Maritan [5] and E. Orlandini [6].

## Keywords

Domain Wall Osmotic Pressure Critical Exponent Universality Class Lattice Gauge Theory## Preview

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## References

- [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]U. Glaus, “Monte Carlo Study of Self Avoiding Surfaces,” J. Stat. Phys. 50, 1141–1166 (1988).CrossRefGoogle Scholar
- [3]A.L. Stella
**“**Self Avoiding Surfaces and Vesicles on a Lattice,”**in:***Complex Systems: Fractals*,*Spin Glasses and Neural Networks***ed. by**G. Parisi, L. Pietronero, and M. Virasoro**Physica A 185**211–221 (1992).Google Scholar - [4]A.L. Stella “Statistical Mechanics of Random Surfaces, Vesicles and Polymers,” Turkish J. Phys. 18, 244–260 (1994).Google Scholar
- [5]A. Maritan, unpublished lecture notes.Google Scholar
- [6]E. Orlandini, unpublished lecture notes.Google Scholar
- [7]A. Maritan and A.L. Stella, “Some Exact Results for Self-Avoiding Random Surfaces,” Nucl. Phys. 280, 561–575 (1987).MathSciNetCrossRefGoogle Scholar
- [8]P.G. de Gennes,
*Scaling Concepts in Polymer Physics*(Cornell University Press, Ithaca, 1979).Google Scholar - [9]P.M. Chaikin and T.C. Lubensky,
*Principles of Condensed Matter Physics*(Cambridge University Press, Cambridge, 1995).Google Scholar - [10]S.G. Whittington, “Statistical Mechanics of Three Dimensional Vesicles,” J. Math. Chem. 14, 103–110 (1993).MathSciNetCrossRefGoogle Scholar
- [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]J.M. Hammersley, “The Number of Polygons on a Lattice,” Proc. Camb. Phil. Soc. 57, 516–523 (1961).MathSciNetMATHCrossRefGoogle Scholar
- [13]A. Baumgartner, “Inflated Vesicles: A Lattice Model,” Physica A 190, 63–74 (1992).CrossRefGoogle Scholar
- [14]A. Baumgartner and A. Romero, “Microcanonical Simulation of Self-Avoiding Surfaces,” Physica A 187, 243–248 (1992).CrossRefGoogle Scholar
- [15]A. Baumgartner, “Phase Transition of Semiflexible Lattice Vesicles,” Physica A 192, 550–561 (1993).CrossRefGoogle Scholar
- [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]T. Sterling and J. Greensite, “Entropy of Self-Avoiding Surfaces on the Lattice,” Phys. Lett. B 121, 345–348 (1983).Google Scholar
- [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]E. Orlandini, Ph.D. Thesis, U. of Bologna, 1993 (unpublished).Google Scholar
- [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]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]B. Baumann and B. Berg, “Non-Trivial Lattice Random Surfaces,” Phys. Lett. 164B, 131–135 (1985).Google Scholar
- [23]A. Maritan and A.L. Stella, “Scaling Behavior of Self-Avoiding Random Surfaces,” Phys. Rev. Lett. 53, 123–126 (1984).CrossRefGoogle Scholar
- [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]S. Redner, “Enumeration Study of Self-Avoiding Random Surfaces,” J. Phys. A 18, L723 L733(1985); 19, 3199 (E) (1986).MathSciNetGoogle Scholar
- [26a]M.E. Cates, “The Fractal Dimension and Connectivity of Random Surfaces,” Phys. Lett. 161B, 363–367 (1985);MathSciNetGoogle Scholar
- [26b]H. Tasaki and T. Hara, “Collapse of Random Surfaces in the Connected Plaquettes Model,” Phys. Lett. 112A 115–118 (1985).MathSciNetGoogle Scholar
- [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
- [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
- [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
- [30]E.J. Janse Van Rensburg, “Crumpling Self-Avoiding Surfaces,” J. Stat. Phys., Vol. 88 - No. 1/2, July 1997.Google Scholar
- [31a]J. Banavar, A. Maritan and A. Stella, “Geometry, Topology, and Universality of Random Surfaces,” Science 252, 825--827 (1991);MathSciNetCrossRefGoogle Scholar
- [31b]“Critical Behavior of Two Dimensional Vesicles in the Deflated Regime,” Phys. Rev. A 43, R5752 R5754 (1991) for the 2D case.Google Scholar
- [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
- [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
- [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