Critical Properties of Some Discrete Random Surface Models

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


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.



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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Copyright information

© Plenum Press, New York 1990

Authors and Affiliations

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

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