Journal of Experimental and Theoretical Physics

, Volume 117, Issue 3, pp 570–578 | Cite as

Finite-temperature perturbation theory for the random directed polymer problem

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


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 〈[δu(X)]2〉, with δu(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 the lower dimensions 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 〈[δu(X)]2〉 ∝ LX 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 〈α2〉 ∝ L, where α = ∂ x u, from which the displacement scaling for the segment X follows naturally, 〈[δu(X)]2〉 ∝ 〈α2X 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.


Thermal Fluctuation Perturbative Expansion Random Potential Short Scale Perturbative Approach 
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Copyright information

© Pleiades Publishing, Inc. 2013

Authors and Affiliations

  • S. E. Korshunov
    • 1
  • V. B. Geshkenbein
    • 2
  • G. Blatter
    • 2
  1. 1.Landau Institute for Theoretical PhysicsRussian Academy of SciencesMoscowRussia
  2. 2.Theoretische PhysikZurichSwitzerland

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