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Critical Phenomena and Fractals with Dimensionality Near 1

  • Y. Gefen
  • A. Aharony
  • B. B. Mandelbrot
Conference paper
Part of the Springer Series in Solid-State Sciences book series (SSSOL, volume 23)

Abstract

This work is concerned with the critical properties of Ising models on self similar fractal lattices [1]. Fractal lattices are not translationally invariant, in contrast to the usual integer dimensional lattices. A fractal’s simplest characteristic is its fractal dimensionality, D, which is ordinarily not an integer. (The ε-expansion arguments are predicated upon the existence of lattices that combine non integer dimensionality with translational invariance, but no such lattice has been actually exhibited). Numerous real physical systems, e.g. polymers and critical percolation clusters, are usefully viewed as self similar fractals [1,2].

Keywords

Ising Model Critical Phenomenon Fractal Dimensionality Sierpinski Gasket Sierpinski Carpet 
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.
    B. B. Mandelbrot, Fractals: Form, Chance and Dimension (W. H. Freeman, San Francisco 1977) and forthcoming sequel.MATHGoogle Scholar
  2. 2.
    H. E. Stanley, R. J. Birgenau, P. J. Reynolds, and J. F. Nicoll, J. Phys. C9, L553 (1976)Google Scholar
  3. 2a.
    H. E. Stanley, R. J. Birgenau, P. J. Reynolds, and J. F. Nicoll B. B. Mandelbrot, Ann. Israel Phys. Soc. 2, 226 (1978)Google Scholar
  4. 2b.
    H. E. Stanley, R. J. Birgenau, P. J. Reynolds, and J. F. Nicoll B. B. Mandelbrot D. Stauffer, Phys. Reports 54, 1 (1979)CrossRefADSGoogle Scholar
  5. 2b.
    H. E. Stanley, R. J. Birgenau, P. J. Reynolds, and J. F. Nicoll B. B. Mandelbrot D. Stauffer, S. Kirkpatrick, Les Houches Summer School on III-Condensed Matter 1978, ed. by R. Balian et al (North Holland, 1979), p. 323.Google Scholar
  6. 3.
    Y. Gefen, B. B. Mandelbrot and A. Aharony, to be published.Google Scholar
  7. 4.
    See e.g. the papers by R. B. Griffiths and by C. J. Thompson in Phase Transitions and Critical Phenomena, edited by C. Domb and M. S. Green (Academic, New York, 1976) vol. 6, p. 357.Google Scholar
  8. 5.
    See also P. Suranyi, Phys. Rev. Lett. 38, 1436 (1977), who used a similar approach to obtain approximate results near TC for the triangular lattice.CrossRefADSGoogle Scholar
  9. 6.
    L. P. Kadanoff, Ann. Phys. (N.Y.) 100, 359 (1976).CrossRefADSGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1981

Authors and Affiliations

  • Y. Gefen
    • 1
  • A. Aharony
    • 1
  • B. B. Mandelbrot
    • 2
  1. 1.Department of Physics and AstronomyTel Aviv UniversityRamat AvivIsrael
  2. 2.IBM Thomas J. Watson Research CenterYorktown HeightsUSA

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