ZhETF, Vol. 144,
No. 3,
p. 653 (September 2013)
(English translation  JETP,
Vol. 117, No. 3,
p. 570,
September 2013
available online at www.springer.com
)
FINITETEMPERATURE PERTURBATION THEORY FOR THE RANDOM DIRECTED POLYMER PROBLEM
Korshunov S.E., Geshkenbein V.B., Blatter G.
Received: April 25, 2013
DOI: 10.7868/S0044451013090150
Dedicated to the memory of Professor Anatoly Larkin} We study the random directed polymer problem  the shortscale 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 shortscale wandering expressed through the displacement correlator , 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 illdefined and the problem turns out to be nonperturbative in low dimension and for an infinitelength 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 depend on L and scale with the square of the segment length X, which differs from a straightforward Larkintype scaling. The result is best understood in terms of a typical squared angle , where , from which the displacement scaling for the segment X follows naturally, . At high temperatures, thermal fluctuations smear the disorder potential and the lowestorder results for disorderinduced 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.

