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ZhETF, Vol. 144, No. 3, p. 653 (September 2013)
(English translation - JETP, Vol. 117, No. 3, p. 570, September 2013 available online at )

Korshunov S.E., Geshkenbein V.B., Blatter G.

Received: April 25, 2013

DOI: 10.7868/S0044451013090150

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Dedicated to the memory of Professor Anatoly Larkin} We study the random directed polymer problem - the short-scale behavior of an elastic string (or polymer) in one transverse dimension subject to a disorder potential and finite temperature fluctuations. We are interested in the polymer short-scale wandering expressed through the displacement correlator \langle [\delta u (X)]^2\rangle, with δ u (X) being the difference in the displacements at two points separated by a distance X. While this object can be calculated at short scales using the perturbation theory in higher dimensions d > 2, this approach becomes ill-defined and the problem turns out to be nonperturbative in low dimension and for an infinite-length polymer. In order to make progress, we redefine the task and analyze the wandering of a string of a finite length L. At zero temperature, we find that the displacement fluctuations \langle [\delta u(X)]^2\rangle \propto L X^2 depend on L and scale with the square of the segment length X, which differs from a straightforward Larkin-type scaling. The result is best understood in terms of a typical squared angle \langle \alpha^2\rangle \propto L, where \alpha = \partial_x u, from which the displacement scaling for the segment X follows naturally, \langle [\delta u (X)]^2\rangle \propto \langle \alpha ^2\rangle X^2. At high temperatures, thermal fluctuations smear the disorder potential and the lowest-order results for disorder-induced fluctuations in both the displacement field and the angle vanish in the thermodynamic limit L → ∞. The calculation up to the second order allows us to identify the regime of validity of the perturbative approach and provides a finite expression for the displacement correlator, albeit depending on the boundary conditions and the location relative to the boundaries.

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