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Fractal structure of zeros in hierarchical models

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Abstract

We consider an Ising and aq-state Potts model on a diamond hierarchical lattice. We give pictures of the distribution of zeros of the partition function in the complex plane of temperatures for several choices ofq. These zeros are just the Julia set corresponding to the renormalization group transformation.

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References

  1. A. N. Berker and S. Ostlund,J. Phys. C12:4961 (1979).

    Google Scholar 

  2. M. Kaufman and R. B. Griffiths,Phys. Rev. B 24:496 (1981).

    Google Scholar 

  3. M. Kaufman and R. B. Griffiths,J. Phys. A15:L239 (1982).

    Google Scholar 

  4. R. B. Griffiths and M. Kaufman,Phys. Rev. B 26:5022 (1982).

    Google Scholar 

  5. S. McKay, A. N. Berker, and S. Kirkpatrick,Phys. Rev. Lett. 48:767 (1982).

    Google Scholar 

  6. N. M. Svrakic, J. Kertész, and W. Selke,J. Phys. A15:L427 (1982).

    Google Scholar 

  7. B. Derrida, J. P. Eckmann, and A. Erzan,J. Phys. A16:893 (1983).

    Google Scholar 

  8. J. R. Melrose,J. Phys. A16:1041 (1983).

    Google Scholar 

  9. A. Erzan,Phys. Lett. A 93A:237 (1983).

    Google Scholar 

  10. A. A. Migdal,Sov. Phys. JETP 42:743 (1976).

    Google Scholar 

  11. L. P. Kadanoff,Ann. Phys. (N.Y.) 100:359 (1976).

    Google Scholar 

  12. P. Fatou,Bull. Math. France 47:161 (1919);48:33, 208 (1920).

    Google Scholar 

  13. G. Julia,J. Math. Pures Appl. 4:47 (1918).

    Google Scholar 

  14. M. J. Feigenbaum,J. Stat. Phys. 19:339 (1978).

    Google Scholar 

  15. B. Derrida, A. Gervois, and Y. Pomeau,J. Phys. A12:269 (1979).

    Google Scholar 

  16. P. Collet and J. P. Eckmann,Iterated Maps on the Interval as Dynamical Systems (Birkhauser, Boston, 1980), and references therein.

    Google Scholar 

  17. B. Mandelbrot,Ann. N.Y. Acad. Sci. 357:249 (1980).

    Google Scholar 

  18. A. Douady, Séminaire Bourbaki No. 599 (1982–1983).

  19. D. Bessis, J. Geronimo, and P. Moussa, to appear inJ. Stat. Phys.

  20. C. N. Yang and T. D. Lee,Phys. Rev. 87:404 (1952); T. D. Lee and C. N. Yang,Phys. Rev. 87:410 (1952).

    Google Scholar 

  21. M. E. Fisher, inLecturers in Theoretical Physics, Vol. VIIC (University of Colorado Press, Boulder, 1965), p. 1.

    Google Scholar 

  22. R. B. Pearson,Phys. Rev. B 26:6285 (1982).

    Google Scholar 

  23. C. Itzykson, R. B. Pearson, and J. B. Zuber,Nucl. Phys. B,220:FS8:415 (1983).

    Google Scholar 

  24. Th. Niemeijer and J. M. J. van Leeuwen, inPhase Transitions and Critical Phenomena, Vol. 6, C. Domb and M. S. Green, eds. (Academic Press, New York, 1976), p. 425.

    Google Scholar 

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Authors and Affiliations

  1. Departement de Physique Generate, Services de Physique Theorique et de Physique du Solide, Commissariat a l'Energie Atomique, Institut de Recherche Fondamentale, Cen-Saclay-91191, Gif-sur-Yvette Cedex, France

    B. Derrida, L. De Seze & C. Itzykson

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  1. B. Derrida
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  2. L. De Seze
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  3. C. Itzykson
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Derrida, B., De Seze, L. & Itzykson, C. Fractal structure of zeros in hierarchical models. J Stat Phys 33, 559–569 (1983). https://doi.org/10.1007/BF01018834

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  • DOI: https://doi.org/10.1007/BF01018834

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