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Free-energy distribution functions for the randomly forced directed polymer | | V.S.Dotsenko
; V.B.Geshkenbein
; D.A.Gorokhov
; G.Blatter
; | | Date: |
6 Jul 2010 | | Abstract: | We study the $1+1$-dimensional random directed polymer problem, i.e., an
elastic string $phi(x)$ subject to a Gaussian random potential $V(phi,x)$ and
confined within a plane. We mainly concentrate on the short-scale and
finite-temperature behavior of this problem described by a short- but
finite-ranged disorder correlator $U(phi)$ and introduce two types of
approximations amenable to exact solutions. Expanding the disorder potential
$V(phi,x) approx V_0(x) + f(x) phi(x)$ at short distances, we study the
random force (or Larkin) problem with $V_0(x) = 0$ as well as the shifted
random force problem including the random offset $V_0(x)$; as such, these
models remain well defined at all scales. Alternatively, we analyze the
harmonic approximation to the correlator $U(phi)$ in a consistent manner.
Using direct averaging as well as the replica technique, we derive the
distribution functions ${cal P}_{L,y}(F)$ and ${cal P}_L(F)$ of free energies
$F$ of a polymer of length $L$ for both fixed ($phi(L) = y$) and free boundary
conditions on the displacement field $phi(x)$ and determine the mean
displacement correlators on the distance $L$. The inconsistencies encountered
in the analysis of the harmonic approximation to the correlator are traced back
to its non-spectral correlator; we discuss how to implement this approximation
in a proper way and present a general criterion for physically admissible
disorder correlators $U(phi)$. | | Source: | arXiv, 1007.0852 | | Services: | Forum | Review | PDF | Favorites | |
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