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Journal of Statistical Physics

, Volume 51, Issue 5–6, pp 817–840 | Cite as

Polymers on disordered trees, spin glasses, and traveling waves

  • B. Derrida
  • H. Spohn
Articles

Abstract

We show that the problem of a directed polymer on a tree with disorder can be reduced to the study of nonlinear equations of reaction-diffusion type. These equations admit traveling wave solutions that move at all possible speeds above a certain minimal speed. The speed of the wavefront is the free energy of the polymer problem and the minimal speed corresponds to a phase transition to a glassy phase similar to the spin-glass phase. Several properties of the polymer problem can be extracted from the correspondence with the traveling wave: probability distribution of the free energy, overlaps, etc.

Key words

Disordered system spin glass freezing transition reaction-diffusion equation 

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References

  1. 1.
    R. Lipowsky and M. E. Fisher,Phys. Rev. Lett. 56:472 (1986).Google Scholar
  2. 2.
    M. E. Fisher,J. Chem. Soc. Faraday Trans. 82:1569 (1986).Google Scholar
  3. 3.
    D. Foster, D. R. Nelson, and M. J. Stephen,Phys. Rev. A 16:732 (1977).Google Scholar
  4. 4.
    H. K. Janssen and B. Schmittman,Z. Phys. B 63:517 (1986).Google Scholar
  5. 5.
    H. van Beijeren, R. Kutner, and H. Spohn,Phys. Rev. Lett. 54:2026 (1985).Google Scholar
  6. 6.
    D. A. Huse, C. L. Henley, and D. S. Fisher,Phys. Rev. Lett. 55:2924 (1985).Google Scholar
  7. 7.
    D. Dhar,Phase Transitions 9:51 (1987).Google Scholar
  8. 8.
    J. Imbrie and T. Spencer, preprint.Google Scholar
  9. 9.
    P. Meakin, P. Ramanlal, L. M. Sander, and R. C. Ball,Phys. Rev. A 34:5091 (1986).Google Scholar
  10. 10.
    D. E. Wolf and J. Kertész,Europhys. Lett. 4:651 (1987).Google Scholar
  11. 11.
    M. Kardar and Y. Zhang,Phys. Rev. Lett. 58:2087 (1987).Google Scholar
  12. 12.
    A. Bovier, J. Fröhlich, and U. Glaus,Phys. Rev. B 34:6409 (1986).Google Scholar
  13. 13.
    M. Kardar, G. Parisi, and Y. Zhang,Phys. Rev. Lett. 56:889 (1986).Google Scholar
  14. 14.
    J. Krug,Phys. Rev. A 36:5465 (1987).Google Scholar
  15. 15.
    A. Kolmogorov, I. Petrovsky, and N. Piscounov,Moscou Univ. Bull. Math. 1:1 (1937).Google Scholar
  16. 16.
    H. P. McKean,Commun. Pure Appl. Math. 28:323 (1975).Google Scholar
  17. 17.
    M. Bramson,Convergence of Solutions of the Kolmogorov Equation to Traveling Waves (Memoirs of the American Mathematical Society, No. 285, 1983).Google Scholar
  18. 18.
    B. Derrida,Phys. Rev. B 24:2613 (1981).Google Scholar
  19. 19.
    B. Derrida and G. Toulouse,J. Phys. Lett. (Paris) 46:223 (1985).Google Scholar
  20. 20.
    M. Mézard, G. Parisi, N. Sourlas, G. Toulouse, and M. Virasoro,J. Phys. (Paris) 45:843 (1984).Google Scholar
  21. 21.
    B. Derrida and H. Flyvbjerg,J. Phys. A 20:5273 (1987).Google Scholar
  22. 22.
    B. Derrida,J. Phys. Lett. 46:401 (1985).Google Scholar
  23. 23.
    B. Derrida and E. Gardner,J. Phys. C 19:5783 (1986).Google Scholar
  24. 24.
    D. Capocaccia, M. Cassandro, and P. Picco,J. Stat. Phys. 46:493 (1987).Google Scholar
  25. 25.
    C. De Dominicis and H. J. Hilhorst,J. Phys. Lett. (Paris) 46:909 (1985).Google Scholar
  26. 26.
    D. Ruelle,Commun. Math. Phys. 108:225 (1987).Google Scholar
  27. 27.
    J. P. Nadal and J. Vannimenus,J. Phys. (Paris) 46:17 (1985), and references therein.Google Scholar
  28. 28.
    A. Bialas and R. Peschanski,Phys. Lett. B, submitted (1988).Google Scholar

Copyright information

© Plenum Publishing Corporation 1988

Authors and Affiliations

  • B. Derrida
    • 1
  • H. Spohn
    • 1
  1. 1.Institute for Theoretical PhysicsUniversity of CaliforniaSanta Barbara

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