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Fractal geometry of critical Potts clusters

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Abstract.

Numerical simulations on the total mass, the numbers of bonds on the hull, external perimeter, singly connected bonds and gates into large fjords of the Fortuin-Kasteleyn clusters for two-dimensional q-state Potts models at criticality are presented. The data are found consistent with the recently derived corrections-to-scaling theory. A new method for thermalization of spin systems is presented. The method allows a speed up of an order of magnetization for large lattices. We also show snapshots of the Potts clusters for different values of q, which clearly illustrate the fact that the clusters become more compact as q increases, and that this affects the fractal dimensions in a monotonic way. However, the approach to the asymptotic region is slow, and the present range of the data does not allow a unique identification of the exact correction exponents.

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References

  1. F.Y. Wu, Rev. Mod. Phys. 54, 235 (1982)

    Article  MathSciNet  Google Scholar 

  2. R.B. Potts, Proc. Camb. Phil. Soc. 48, 106 (1952)

    MathSciNet  MATH  Google Scholar 

  3. C.M. Fortuin, P.W. Kasteleyn, Physica (Utrecht) 57, 536 (1972)

    Article  Google Scholar 

  4. R. Swendsen, J. Wang, Phys. Rev. Lett. 58, 86 (1987)

    Article  Google Scholar 

  5. U. Wolf, Phys. Rev. Lett. 62, 361 (1989)

    Article  Google Scholar 

  6. E. Ising, Z. Phys. 21, 613 (1925)

    Google Scholar 

  7. S. Alexander, Phys. Lett. A 54, 353 (1975)

    Article  Google Scholar 

  8. M. Bretz, Phys. Rev. Lett. 38, 501 (1977)

    Article  Google Scholar 

  9. E. Domany, M. Schick, J.S. Walker, Phys. Rev. Lett. 38, 1148 (1977)

    Article  Google Scholar 

  10. D. Stauffer, A. Aharony, Introduction to Percolation Theory, revised 2nd edition ed. (Taylor and Francis, Burgess Science Press, Basingstoke, London, 1994)

  11. A. Coniglio, J. Phys. A 15, 3829 (1982)

    Article  MathSciNet  Google Scholar 

  12. A. Coniglio, Phys. Rev. Lett. 46, 250 (1981)

    Article  Google Scholar 

  13. R. Pike, H.E. Stanley, J. Phys. A 14, L169 (1981)

  14. B.B. Mandelbrot, The Fractal Geometry of Nature (W.H. Freedman and Company, New York, 1977)

  15. R.F. Voss, J. Phys. A 17, L373 (1984)

  16. A. Bunde, J.F. Gouyet, J. Phys. A 18, L285 (1985)

  17. T. Grossman, A. Aharony, J. Phys. A 19, L745 (1986)

  18. M. Aizenman, B. Duplantier, A. Aharony, Phys. Rev. Lett. 83, 1359 (1999)

    Article  Google Scholar 

  19. H. Saleur, B. Duplantier, Phys. Rev. Lett. 58, 2325 (1987)

    Article  MathSciNet  Google Scholar 

  20. B. Duplantier, Phys. Rev. Lett. 84, 1363 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  21. A. Aharony, J. Asikainen, in Scaling and Disordered Systems, edited by F. Family, M. Daoud, H.J. Herrmann, H.E. Stanley (World Scientific, Singapore, 2002)

  22. N. Metropolis , J. Chem. Phys 21, 1087 (1953)

    Google Scholar 

  23. J.L. Cardy, M. Nauenberg, D.J. Scalapino, Phys. Rev. B 22, 2560 (1980)

    Article  MathSciNet  Google Scholar 

  24. C. Vanderzande, J.F. Marko, J. Phys. A 26, 7391 (1993)

    MathSciNet  Google Scholar 

  25. A. Aharony, G. Ahlers, Phys. Rev. Lett. 44, 782 (1980)

    Article  Google Scholar 

  26. M.P.M. den Nijs, Phys. Rev. B 27, 1674 (1983)

    Article  Google Scholar 

  27. Y. Gefen, B.B. Mandelbrot, A. Aharony, A. Kapitulnik, J. Stat. Phys. 36, 827 (1984)

    MathSciNet  Google Scholar 

  28. M.E.J. Newman, R.M. Ziff, Phys. Rev. Lett. 85, 4104 (2000)

    Article  Google Scholar 

  29. J. Hoshen, R. Kopelman, Phys. Rev. B 14, 3438 (1976)

    Google Scholar 

  30. W.H. Press, S.A. Teukolsky, W.T.V.B.P. Flannery, Numerical Recipes in C: The Art of Scientific Computing (Cambridge University Press, Great Britain, Cambridge, 1992)

  31. A. Coniglio, Phys. Rev. Lett. 62, 3054 (1989)

    MathSciNet  Google Scholar 

Download references

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Authors and Affiliations

  1. Helsinki Institute of Physics and Laboratory of Physics, Helsinki University of Technology, Finland

    J. Asikainen

  2. Raymond and Beverly Sackler Faculty of Exact Sciences, School of Physics and Astronomy Tel Aviv University, 69978, Tel Aviv, Israel

    A. Aharony & J.-P. Hovi

  3. Department of Mathematics, Yale University, CT 06520-8283, New Haven, USA

    B. B. Mandelbrot, E. Rausch & J.-P. Hovi

Authors
  1. J. Asikainen
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  2. A. Aharony
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  3. B. B. Mandelbrot
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  4. E. Rausch
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  5. J.-P. Hovi
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Correspondence to J. Asikainen.

Additional information

Received: 2 June 2003, Published online: 9 September 2003

PACS:

05.50.+q Lattice theory and statistics (Ising, Potts, etc.) - 05.45.Df Fractals - 75.10.-b General theory and models of magnetic ordering - 75.40.Cx Static properties (order parameter, static susceptibility, heat capacities, critical exponents, etc.)

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Asikainen, J., Aharony, A., Mandelbrot, B.B. et al. Fractal geometry of critical Potts clusters. Eur. Phys. J. B 34, 479–487 (2003). https://doi.org/10.1140/epjb/e2003-00247-7

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  • DOI: https://doi.org/10.1140/epjb/e2003-00247-7

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