Abstract
We prove the moments of the directed polymer partition function GZ, using an exact position space renormalization group scheme on a hierarchical lattice. After sufficient iteration the characteristic functionf(n)=ln〈GZn〉 of the probability ℘(Z) converges to a stable limitf *(n). For smalln the limiting behavior is independent of the initial distribution, while for largen,f *(n) is completely determined by it and is thus nonuniversal. There is a smooth crossover between the two regimes for small effective dimensions, and the nonlinear behavior of the small moments can be used to extract information on the universal scaling properties of the distribution. For large effective dimensions there is a sharp transition between the two regimes, and analytical continuation from integer moments ton→0 is not possible. Replica arguments can account for most features of the observed results.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
M. Kardar and Y.-C. Zhang,Phys. Rev. Lett. 58:2087 (1987).
M. Kardar, G. Parisi, and Y.-C. Zhang,Phys. Rev. Lett. 56:889 (1986); E. Medina, T. Hwa, M. Kardar, and Y.-C. Zhang,Phys. Rev. B 39:3053 (1989).
V. L. Nguyen, B. Z. Spivak, and B. I. Shklovskii,Pis'ma Zh. Eksp. Teor. Fiz. 41:35 (1985) [JETP Lett. 41:42 (1985)];Zh. Eksp. Teor. Fiz. 89:11 (1985).
E. Medina, M. Kardar, Y. Shapir, and X.-R. Wang,Phys. Rev. Lett. 62:941 (1989).
D. Huse and C. L. Henley,Phys. Rev. Lett. 54:2708 (1985).
J. Z. Imbrie and T. Spencer,J. Stat. Phys. 52:609 (1988).
M. Mezard and G. Parisi,J. Phys. A 25:4521 (1992).
T. Blum and Y. Y. Goldschmidt,J. Phys. A 25:6517(1992); Y. Y. Goldschmidt,J. Phys. I (Paris)2:1607 (1992).
M. Kardar,Nucl. Phys. B 290 [FS20]:582 (1987).
Y.-C. Zhang,Phys. Rev. Lett. 62:979 (1989);Europhys. Lett. 9:113 (1989);J. Stat. Phys. 57:1123 (1989).
A. Crisanti, G. Paladin, H.-J. Sommers, and A. Vulpiani,J. Phys. I (Paris)2:1325 (1992).
T. Halpin-Healy,Phys. Rev. A 44:3415 (1991).
J. Krug, P. Meakin, and T. Halpn-Healy,Phys. Rev. A 45:638 (1992).
J. M. Kim, M. A. Moore, and A. J. Bray,Phys. Rev. A 44:2345 (1991).
A. N. Berker and S. Ostlund,J. Phys. C 12:4961 (1979).
B. Derrida and R. B. Griffiths,Europhys. Lett. 8:111 (1989).
J. Cook and B. Derrida,J. Stat. Phys. 57:89 (1989).
S. Roux, A. Hansen, L. R. da Silva, L. S. Lucena, and R. B. Pandey,J. Stat. Phys. 65:183 (1991).
Ł. Balents and M. Kardar,J. Stat. Phys. 67:1 (1992).
T. Blum, Y. Shapir, and D. Koltun,J. Phys. I (Paris)1:613 (1991).
M. R. Evans and B. Derrida,J. Stat. Phys. 69:427 (1992).
E. Medina and M. Kardar,Phys. Rev. B 46:9984 (1992).
E. B. Kolomeisky,JETP Lett. 54:57 (1992).
M. P. Gelfand,Physica A 177:67 (1991).
B. L. Altshuler, V. E. Kravtsov, and I. V. Lerner,JETP Lett. 43:441 (1986);Sov. Phys. JETP 64:1352 (1986).
B. Shapiro,Phys. Rev. Lett. 65:1510 (1990).
N. I. Akhiezer,The Classical Moment Problem (Oliver and Boyd, London, 1965).
M. Shreiber and H. Grussbach,Phys. Rev. Lett. 67:607 (1991).
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Medina, E., Kardar, M. Nonuniversality and analytical continuation in moments of directed polymers on hierarchical lattices. J Stat Phys 71, 967–980 (1993). https://doi.org/10.1007/BF01049956
Received:
Issue Date:
DOI: https://doi.org/10.1007/BF01049956