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Recent progress and current puzzles in percolation

Conference Lectures

Part of the Lecture Notes in Mathematics book series (LNM,volume 1035)

Abstract

The basic physical phenomena of percolation are reviewed within the context of the modern theory of critical phenomena. The connection between percolation and the Potts model, a statistical mechanical model of ferromagnetism, is discussed. Recent advances in calculating critical exponents by position-space renormalization group methods are also described. Several open questions are also raised, including the nature of cluster structure and transport near the percolation threshold, and the anomalous geometrical properties of self-similar structures.

Keywords

  • Critical Exponent
  • Percolation Threshold
  • Cluster Structure
  • Critical Phenomenon
  • Random Medium

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. Alexander, S. and Orbach, R. 1982, J. de Physique 43, L625.

    Google Scholar 

  2. Bernasconi, J., 1978, Phys. Rev. B. 18, 2185.

    CrossRef  ADS  MathSciNet  Google Scholar 

  3. Broadbent, S.R. and Hammersley, J.M., 1957, Proc. Camb. Philos. Soc. 53, 624.

    CrossRef  ADS  MathSciNet  Google Scholar 

  4. Coniglio, A., 1981a, Phys. Rev. Lett. 46, 250.

    CrossRef  ADS  Google Scholar 

  5. Coniglio, A., 1981b, in Lecture Notes in Physics, Disordered Systems and Localization, Proc. Rome Conf., 1981, ed. C. Castllani, C. Di Castro and L. Peliti (Springer: Berlin).

    Google Scholar 

  6. Coniglio, A., 1982, J. Phys. A. 15, 3824.

    ADS  Google Scholar 

  7. Derrida, B. and Vannimenus, J., 1982, J. Phys. A. 15, L557.

    Google Scholar 

  8. Essam, J.W., 1980, Rep. Prog. Phys. 43, 833.

    CrossRef  ADS  MathSciNet  Google Scholar 

  9. Family, F., 1980, J. Phys. A. 13, L325.

    Google Scholar 

  10. Fogelholm, R., 1980, J. Phys. C. 13, L571.

    Google Scholar 

  11. Fortuin, C.M. and Kasteleyn, P.W., 1972, Physica 57, 536.

    CrossRef  ADS  MathSciNet  Google Scholar 

  12. Frank, D.J. and Lobb, C.J., 1983, unpublished.

    Google Scholar 

  13. Gefen, Y., Aharony, A., and Alexander, S., 1983, Phys. Rev. Lett 50, 77.

    CrossRef  ADS  Google Scholar 

  14. Gefen, Y., Aharony, A., Mandelbrot, B.B. and Kirkpatrick, 1981. Phys. Rev. Lett. 47, 1771.

    CrossRef  ADS  MathSciNet  Google Scholar 

  15. de Gennes, P.G., 1976, La Recherche 7, 919.

    Google Scholar 

  16. Kasteleyn, P.W. and Fortuin, C.M. 1969, J. Phys. Soc. Japan (suppl) 26, 11.

    ADS  Google Scholar 

  17. Lobb, C.J. and Frank, D.J., 1979, J. Phys. C. 9, L827.

    Google Scholar 

  18. Ma, S.-K., 1976, Modern Theory of Critical Phenomena (Benjamin: Reading MA).

    Google Scholar 

  19. den Nijs, M.P.M., 1979, J. Phys. A. 12, 1857.

    CrossRef  ADS  Google Scholar 

  20. Pearson, R.B., 1980, Phys. Rev. B. 22, 2579.

    CrossRef  ADS  Google Scholar 

  21. Pfeuty, P. and Toulouse, G., 1977, Introduction to the Renormalization Group and to Critical Phenomena (Wiley: London).

    Google Scholar 

  22. Pike, R. and Stanley, H.E., 1981, J. Phys. A 14, L169.

    Google Scholar 

  23. Rammal, R. and Toulouse, G., 1983, J. de Physique 44, L13.

    Google Scholar 

  24. Render, S., 1982, Phys. Rev. B 24, 3242.

    ADS  Google Scholar 

  25. Redner, S., 1983, in Percolation Structure and Processes, ed. G. Deutscher, R. Zallen, and J. Adler. Annals of the Israel Physical Society, vol. 5 (Hilger: Bristol).

    Google Scholar 

  26. Redner, S. and Reynolds, P.J., 1981, J. Phys. A 14, 2679.

    CrossRef  ADS  MathSciNet  Google Scholar 

  27. Reynolds, P.J., Klein, W. and Stanley, H.E., 1977, J. Phys. C 10, L167.

    Google Scholar 

  28. Reynolds, P.J., Stanley, H.E., and Klein, W., 1980, Phys. Rev. B 21, 1223.

    CrossRef  ADS  Google Scholar 

  29. Skal, A.S. and Shklovskii, B.I., 1975, Sov. Phys. Semicond. 8, 1029.

    Google Scholar 

  30. Stanley, H.E., 1981, in Lecture Notes in Physics, Disordered Systems and Localization, Proc. Rome Conf. 1981, ed C. Castellani, C. Di Castro and L. Peliti (Springer: Berlin).

    Google Scholar 

  31. Stanley, H.E., and Coniglio, A., in Percolation Structure and Processes, ed. G. Deutscher, R. Zallen, and J. Adler. Annals of the Israel Physical Society, vol 5 (Hilger: Bristol).

    Google Scholar 

  32. Stanley, H.E., Reynolds, P.J., Redner, S., and Family, F., 1982 in "Real-Space Renormalization" Topics in Current Physics 30, chap 7 (Springer: Berlin).

    CrossRef  Google Scholar 

  33. Stauffer, D., 1979, Phys. Rept. 54, 1

    CrossRef  ADS  Google Scholar 

  34. Stephen, M.J., 1976, Phys. Lett. 56A, 149.

    CrossRef  ADS  Google Scholar 

  35. Wu, F.Y., 1982. Rev. Mod. Phys. 54, 235.

    CrossRef  ADS  Google Scholar 

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© 1983 Springer-Verlag

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Redner, S. (1983). Recent progress and current puzzles in percolation. In: Hughes, B.D., Ninham, B.W. (eds) The Mathematics and Physics of Disordered Media: Percolation, Random Walk, Modeling, and Simulation. Lecture Notes in Mathematics, vol 1035. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0073260

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  • DOI: https://doi.org/10.1007/BFb0073260

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