Minimal path on the hierarchical diamond lattice

Abstract

We consider the minimal paths on a hierarchical diamond lattice, where bonds are assigned a random weight. Depending on the initial distribution of weights, we find all possible asymptotic scaling properties. The different cases found are the small-disorder case, the analog of Lévy's distributions with a power-law decay at-∞, and finally a limit of large disorder which can be identified as a percolation problem. The asymptotic shape of the stable distributions of weights of the minimal path are obtained, as well as their scaling properties. As a side result, we obtain the asymptotic form of the distribution of effective percolation thresholds for finite-size hierarchical lattices.

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References

  1. 1.

    M. Kardar, G. Parisi, and Y. C. Zhang,Phys. Rev. Lett. 56:889 (1986).

    Google Scholar 

  2. 2.

    M. Kardar and Y. C. Zhang,Phys. Rev. Lett. 58:2087 (1987).

    Google Scholar 

  3. 3.

    S. Roux, A. Hansen, and E. Guyon,J. Phys. (Paris)48:2125 (1987).

    Google Scholar 

  4. 4.

    D. A. Huse and C. L. Henley,Phys. Rev. Lett. 54:2708 (1985); M. Kardar,Phys. Rev. Lett. 55:2923 (1985); D. A. Huse, C. L. Henley, and D. S. Fisher,Phys. Rev. Lett. 55:2924 (1985).

    Google Scholar 

  5. 5.

    D. E. Wolf and J. Kertész,Europhys. Lett. 4:651 (1987).

    Google Scholar 

  6. 6.

    A. J. McKane and M. A. Moore,Phys. Rev. Lett. 60:527 (1988).

    Google Scholar 

  7. 7.

    J. M. Kim and J. M. Kosterlitz,Phys. Rev. Lett. 62:2289 (1989); D. E. Wolf and J. Kertész,Phys. Rev. Lett. 63:1191 (1989); J. M. Kim and J. M. Kosterlitz,Phys. Rev. Lett. 63:1192 (1989).

    Google Scholar 

  8. 8.

    B. Derrida and R. B. Griffiths,Europhys. Lett. 8:111 (1989).

    Google Scholar 

  9. 9.

    T. Halpin-Healy,Phys. Rev. A 42:711 (1990);Phys. Rev. Lett. 63:917 (1989).

    Google Scholar 

  10. 10.

    D. Stauffer,Introduction to Percolation Theory (Taylor and Francis, London, 1987).

    Google Scholar 

  11. 11.

    J. P. Bouchaud and A. Georges,Phys. Rep., to appear.

  12. 12.

    L. Niemeyer, L. Pietronero, and H. J. Weismann,Phys. Rev. Lett. 52:286 (1984).

    Google Scholar 

  13. 13.

    S. Roux, A. Hansen, and E. L. Hinrichsen,J. Phys. A 24:L295 (1991).

    Google Scholar 

  14. 14.

    Y. C. Zhang,J. Phys. (Paris)51:2129 (1990).

    Google Scholar 

  15. 15.

    J. G. Amar and F. Family,J. Phys. A 24:L79 (1991).

    Google Scholar 

  16. 16.

    Y. C. Zhang, Preprint.

  17. 17.

    J. Krug, Preprint.

  18. 18.

    B. Derrida,Physica A 163:71 (1990).

    Google Scholar 

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Roux, S., Hansen, A., da Silva, L.R. et al. Minimal path on the hierarchical diamond lattice. J Stat Phys 65, 183–204 (1991). https://doi.org/10.1007/BF01329855

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Key words

  • Minimal path
  • directed polymer
  • hierarchical lattice
  • percolation