J. Phys. I France
Volume 3, Numéro 11, November 1993
Page(s) 2179 - 2198
DOI: 10.1051/jp1:1993240
J. Phys. I France 3 (1993) 2179-2198

Directed polymers in the presence of columnar disorder

Joachim Krug1 and Timothy Halpin-Healy2

1  Institut für Festkörperforschung, Forschungszentrum Jülich, P.O. Box 1913, D-52425 Jülich, Germany
2  Physics Departement, Barnard College, Columbia University, New York, NY 10027-6598, U.S.A.

(Received 10 March 1993, revised 28 June 1993, accepted 21 July 1993)

We consider directed polymers in a random landscape that is completely correlated in the time direction. This problem is closely related to diffusion-reproduction processes and undirected Gaussian polymers in a disordered environment. In contrast to the case of uncorrelated disorder, we find the behavior to be very different at zero temperature, where the scaling exponents depend on the details of the random energy distribution, and at finite temperature, where the transverse wandering is subballistic, $x \sim t/(\log t)^{\gamma}$ with $\gamma = 1 + 2/d$ for bounded distributions in d + 1 dimensions. Numerically, these strong logarithmic corrections give rise to apparently nontrivial effective exponents. Our analytic results are based on appropriate Flory expressions for the (free) energy at T = 0 and T > 0. Some universal statistical properties of the evolutionary hopping of the optimal path are also derived.

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