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Critical Properties of Some Discrete Random Surface Models

  • Bergfinnur Durhuus
Part of the NATO ASI Series book series (NSSB, volume 234)

Abstract

Detailed insight into the statistical mechanics of random surfaces, and, in particular, the universality properties of such systems, is important for understanding a variety of phenomena in both condensed matter and high energy physics, of which we may note the following (see also refs. [1,2,3,4]):
  1. 1)

    Crystal growth and properties of crystalline surfaces.

     
  2. 2)

    Interfaces separating different phases of a system, appearing e.g. in wetting phenomena or as domain walls in ferromagnets.

     
  3. 3)

    Lipid bilayers, which are also interesting because of their similarity with certain biological membranes. These may be characterized as fluid membranes, meaning that their molecular constituents behave as a two-dimensional fluid.

     
  4. 4)

    Micro-emulsions, f.ex. mixtures of oil and water with a surfactant added. An essential feature of these is a very tiny surface tension, and the sizes of oil/water regions are of the order of 100 Å, i.e. much smaller than in ordinary emulsions where they are typically around 10−5 m.

     
  5. 5)

    Membranes with crystalline (or hexatic) order, as opposed to fluid membranes.

     
  6. 6)

    Random surface expansions in lattice gauge theory. Examples of such are strong conpling expansions and large -N expansions.

     
  7. 7)
    Effective theory for QCD flux tubes, or hadronic strings. Here the surfaces appear as world sheets of propagating strings. Using Feynman’s approach to quantum mechanics one is lead to consider expressions of the form
    $$\int{DS{{e}^{i/hA\left( S \right)}}}$$
    for transition amplitudes between string states. Here A(S) is an action attributed to each surface S and DS is a measure on a suitable set of surfaces. One way to give a rigorous meaning to such an integral is by constructing continuum limits of corresponding well defined discrete expressions, as we shall discuss in the following.
     
  8. 8)

    (Super-) string theory, viewed as a fundamental theory of elementary particles.

     
  9. 9)

    Two-dimensional quantum gravity.

     

Keywords

Random Walk Critical Exponent Continuum Limit Boundary Component Critical Index 
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.

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References

  1. 1.
    H.J. Leamy, G.H. Gilmer, K.A. Jackson, in: “Surface Physics of Crystalline Materials”, J.M. Blakely (ed.), New York, Academic Press, 1976.Google Scholar
  2. 2.
    J. Fröhlich, C. Pfister, T. Spencer, in: “Springer Lecture Notes in Physics” 173, Berlin-Heidelberg-New York, Springer-Verlag, 1982.Google Scholar
  3. 3.
    P.G. De Gennes, C. Taupin “J. Phys. Chem.,” 86 1982, p. 2294.CrossRefGoogle Scholar
  4. 4.
    W. Helfrich “Z. Naturforsch.,” 28c, 1973, p. 693.MathSciNetGoogle Scholar
  5. 5.
    J. Glimm, A. Jaffe, “Quantum Physics,” New York-Heidelberg-Berlin, Springer Verlag, 1981.MATHGoogle Scholar
  6. 6.
    D. Weingarten, “Phys.Lett.” 90B, 1980, p. 280.CrossRefGoogle Scholar
  7. 7.
    B. Durhuus, J. Fröhlich, T. Jónsson, “Nucl. Phys.” B225 [FS9], 1983, p. 185.ADSCrossRefGoogle Scholar
  8. 8.
    D. Ruelle, “Statistical Mechanics, Rigorous Results,” New York, Benjamin 1969.MATHGoogle Scholar
  9. 9.
    T. Eguchi, H. Kawai, “Phys. Lett.” 114B, 1982, p. 247.MathSciNetCrossRefGoogle Scholar
  10. 10.
    B. Durhuus, J. Fröhlich, T. Jónsson, “Phys. Lett.” 137B, 1984, p. 93.MathSciNetCrossRefGoogle Scholar
  11. 11.
    B. Durhuus, J. Fröhlich, T. Jónsson “Nucl. Phys.” B240 [FS12], 1984, p. 453.ADSCrossRefGoogle Scholar
  12. 12.
    H. Kawai, Y. Okamoto, “Phys.Lett. ” 130B, 1983, p. 415.ADSGoogle Scholar
  13. 13.
    J. Drouffe, G. Parisi, N. Sourlas, “Nucl. Phys.” B161, 1980, p397.ADSCrossRefGoogle Scholar
  14. 14.
    T. Jónsson, “Comm.Math.Phys.” 106, 1986, p. 679.MathSciNetADSMATHCrossRefGoogle Scholar
  15. 15.
    W. Helfrich, “J. Physique 46, 1985, p.1263; L. Peliti, S. L.ibler, ” Phys.Rev.Lett.“ 54, 1985, p.1690; D. Förster, ”Phys. Lett.“ 114A, 1986, p.115; A.M. Polyakov, ”Nucl. Phys.“ B268, 1986, p$1406; H. Kleinert, ”Phys. Lett.“ 174B, 1986, p. 335.Google Scholar
  16. 16.
    B. Durhuus, T. Jónsson, “Phys. Lett.” 180B, 1986, p. 385.MathSciNetCrossRefGoogle Scholar
  17. 17.
    J. Ambjorn, B. Durhuus, J. Fröhlich, T. Jónsson, in: preparation.Google Scholar
  18. 18.
    Y. Kantor, D.R. Nelson, “Phys.Rev.Lett.” 58, 1987, p.2774; D.R. Nelson, L. Peliti, “J. Physique” 48, 1987, p.1085; F. David, E. Guitter, “Europhys. Lett.” 5, 1988, p.709; M. Baig, D. Espriu, J.F. Wheater, Univ. of Oxford preprint, 1988; J. Ambjorn, B. Durhuus, T. Jónsson, to appear in Nucl. Phys. B.Google Scholar
  19. 19.
    J. Ambjorn, B. Durhuus, T. Jónsson, “Journ. Phys.” A 21, 1988, p. 981.ADSGoogle Scholar
  20. 20.
    A.M. Polyakov, “Phys. Lett.” 103B, 1981, p. 207.MathSciNetCrossRefGoogle Scholar
  21. 21.
    W.T. Tutte, “Can. J. Math.” 14, 1962, p. 21.MathSciNetMATHCrossRefGoogle Scholar
  22. 22.
    J. Ambjorn, B. Durhuus, J. Fröhlich, “Nucl. Phys.” B257, 1985, p. 433.MathSciNetADSCrossRefGoogle Scholar
  23. 23.
    A. Billoire, D. Gross, E. Marinari, “Phys. Lett.” 139B, 1984, p. 75.MathSciNetCrossRefGoogle Scholar
  24. 24.
    J. Ambjorn, B. Durhuus, J. `Fröhlich, T. Jónsson, Nucl. Phys.“ B290 [FS20], 1987, p. 480.MathSciNetADSCrossRefGoogle Scholar
  25. 25.
    J. Fröhlich, in: “Recent Developments in Quantum Field Theory”, eds. J. Ambjorn, B. Durhuus, J.L. Petersen, Amsterdam-Oxford-New York-Tokyo, North-Holland, 1985.Google Scholar
  26. 26.
    J. Ambjorn, B. Durhuus, “Phys. Lett.” 188B, 1987, p. 253.MathSciNetCrossRefGoogle Scholar
  27. 27.
    F. David, “Nucl. Phys.” B257 [FS14], 1985, p. 543.Google Scholar
  28. 28.
    V. Kazakov, I. Kostov, A.A. Migdal, “Phys. Lett.” 157B, 1985, p. 295.MathSciNetCrossRefGoogle Scholar
  29. 29.
    N. Nakanishi, “Graph Theory and Feynman Integrals”, Gordon and Breach 1971.Google Scholar
  30. 30.
    J. Ambjorn, B. Durhuus, J. Fröhlich, P. Orland, “Nucl. Phys.” B270 [FS16], 1986, p. 457.Google Scholar
  31. 31.
    J. Ambjorn, B. Durhuus, J. Fröhlich, “Nucl. Phys.” B275 [FS17], 1986, p. 161.Google Scholar
  32. 32.
    D.V. Boulatov, V. Kazakov, I. Kostov, A.A. Migdal, “Nucl. Phys.” B275, [FS17] p.641.Google Scholar
  33. 33.
    F. David, J. Jurkiewicz, A. Krzywicki, B. Peterson. “Nucl. Phys.” B290, [FS20] 1987, p. 218.ADSCrossRefGoogle Scholar
  34. 34.
    A. Billoire, F. David, “Nucl. Phys.” B275, 1986, p. 617.MathSciNetADSCrossRefGoogle Scholar
  35. 35.
    J. Ambjorn, Ph. De Forcrand, F. Koukiou, D. Petritis, “Phys. Lett.” 197B, 1987, p. 548.MathSciNetCrossRefGoogle Scholar
  36. 36.
    F. David, “Nucl. Phys.” B257, 1985, p. 45.ADSCrossRefGoogle Scholar
  37. 37.
    V. Kazakov, A.A. Migdal, Niels Bohr Institut preprint NBI-HE-88–28, 1988.Google Scholar
  38. 38.
    A.M. Polyakov, “Mod.Phys. Lett.” A2, 1987, p.893; V. Knizhnik, A.M. Polyakov, A.B. Zamolodchikov, “Mod. Phys. Lett.” A3, 1988, p. 819.Google Scholar
  39. 39.
    F. David, Saclay preprint SPhT/88–132, 1988.Google Scholar
  40. 40.
    M.E. Cates communication through F. David.Google Scholar
  41. 41.
    F. David, E. Guitter, “Nucl. Phys.” B295 [FS21], 1988, p. 332.Google Scholar
  42. 42.
    R. Giles, C. Thorn, “Phys. Rev.” D16, 1977, p. 366.ADSGoogle Scholar
  43. 43.
    P. Orland, “Nucl. Phys” B278, 1986, p. 790.Google Scholar
  44. 44.
    S. Mandelstam, “Phys. Rep.” 13, 1974, p. 259.ADSCrossRefGoogle Scholar
  45. 45.
    B. Durhuus, J. Fröhlich, T. Jónsson, “Nucl. Phys.” B257 [FS14], 1985, p. 779.ADSCrossRefGoogle Scholar
  46. 46.
    J. Fröhlich, T. Spencer, “Comm. Math. Phys.” 81, 1981, p. 527.MathSciNetADSCrossRefGoogle Scholar
  47. 47.
    D.B. Abraham, J.T. Chayes, L. Chayes, “Phys. Rev.” D30, 1984, p.841, “Comm. Math. Phys.” 96, 1984, p. 439.Google Scholar
  48. 48.
    R. Schrader, “Comm. Math. Phys.” 102, 1985, p. 31.MathSciNetADSCrossRefGoogle Scholar
  49. 49.
    N. Christ, R. Friedberg, T.D. Lee, “ Nucl. Phys.” B202, 1982, p.89, B210 [FS6] 1982, pp. 310, 317.Google Scholar
  50. 50.
    M. Bander, C. Itzykson, in: “Springer Lecture Notes in Physics” 226, Berlin, Springer Verlag, 1985.Google Scholar
  51. 51.
    J.M. Drouffe, H. Kluberg-Stern, “Nucl. Phys.” B260, 1985, p. 253.MathSciNetADSCrossRefGoogle Scholar
  52. 52.
    M.A. Bershadski, A.A. Migdal, “Phys. Lett.” 174B, 1986, p. 393.MathSciNetCrossRefGoogle Scholar
  53. 53.
    V. Kazakov, “Phys. Lett.” 119A, 1986, p. 140.MathSciNetCrossRefGoogle Scholar
  54. 54.
    B. Duplantier, I. Kostov, Saclay preprint SPhT/88–119, 1988.Google Scholar
  55. 55.
    B. Baumann, B Berg, “Phys. Lett.” B164, 1985, p.131; B. Baumann, “Nucl. Phys.” B285 [FS19], 1987, p.391; B. Baumann, B. Berg, G. Münster, DESY preprint, 1988.Google Scholar

Copyright information

© Plenum Press, New York 1990

Authors and Affiliations

  • Bergfinnur Durhuus
    • 1
  1. 1.Mathematics InstituteUniversity of CopenhagenCopenhagen ØDenmark

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