Free-energy distribution functions for the randomly forced directed polymer
arXiv:1007.0852 · doi:10.1103/PhysRevB.82.174201
Abstract
We study the $1+1$-dimensional random directed polymer problem, i.e., an elastic string $Ï(x)$ subject to a Gaussian random potential $V(Ï,x)$ 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 $U(Ï)$ and introduce two types of approximations amenable to exact solutions. Expanding the disorder potential $V(Ï,x) \approx V_0(x) + f(x) Ï(x)$ 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 $U(Ï)$ 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 ($Ï(L) = y$) and free boundary conditions on the displacement field $Ï(x)$ 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 $U(Ï)$.
16 pages, 5 figures