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.
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References
R. Lipowsky and M. E. Fisher,Phys. Rev. Lett. 56:472 (1986).
M. E. Fisher,J. Chem. Soc. Faraday Trans. 82:1569 (1986).
D. Foster, D. R. Nelson, and M. J. Stephen,Phys. Rev. A 16:732 (1977).
H. K. Janssen and B. Schmittman,Z. Phys. B 63:517 (1986).
H. van Beijeren, R. Kutner, and H. Spohn,Phys. Rev. Lett. 54:2026 (1985).
D. A. Huse, C. L. Henley, and D. S. Fisher,Phys. Rev. Lett. 55:2924 (1985).
D. Dhar,Phase Transitions 9:51 (1987).
J. Imbrie and T. Spencer, preprint.
P. Meakin, P. Ramanlal, L. M. Sander, and R. C. Ball,Phys. Rev. A 34:5091 (1986).
D. E. Wolf and J. Kertész,Europhys. Lett. 4:651 (1987).
M. Kardar and Y. Zhang,Phys. Rev. Lett. 58:2087 (1987).
A. Bovier, J. Fröhlich, and U. Glaus,Phys. Rev. B 34:6409 (1986).
M. Kardar, G. Parisi, and Y. Zhang,Phys. Rev. Lett. 56:889 (1986).
J. Krug,Phys. Rev. A 36:5465 (1987).
A. Kolmogorov, I. Petrovsky, and N. Piscounov,Moscou Univ. Bull. Math. 1:1 (1937).
H. P. McKean,Commun. Pure Appl. Math. 28:323 (1975).
M. Bramson,Convergence of Solutions of the Kolmogorov Equation to Traveling Waves (Memoirs of the American Mathematical Society, No. 285, 1983).
B. Derrida,Phys. Rev. B 24:2613 (1981).
B. Derrida and G. Toulouse,J. Phys. Lett. (Paris) 46:223 (1985).
M. Mézard, G. Parisi, N. Sourlas, G. Toulouse, and M. Virasoro,J. Phys. (Paris) 45:843 (1984).
B. Derrida and H. Flyvbjerg,J. Phys. A 20:5273 (1987).
B. Derrida,J. Phys. Lett. 46:401 (1985).
B. Derrida and E. Gardner,J. Phys. C 19:5783 (1986).
D. Capocaccia, M. Cassandro, and P. Picco,J. Stat. Phys. 46:493 (1987).
C. De Dominicis and H. J. Hilhorst,J. Phys. Lett. (Paris) 46:909 (1985).
D. Ruelle,Commun. Math. Phys. 108:225 (1987).
J. P. Nadal and J. Vannimenus,J. Phys. (Paris) 46:17 (1985), and references therein.
A. Bialas and R. Peschanski,Phys. Lett. B, submitted (1988).
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Derrida, B., Spohn, H. Polymers on disordered trees, spin glasses, and traveling waves. J Stat Phys 51, 817–840 (1988). https://doi.org/10.1007/BF01014886
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DOI: https://doi.org/10.1007/BF01014886