Free-energy distribution functions for the randomly forced directed polymer

Abstract

We study the $1+1$-dimensional random directed polymer problem, i.e., an elastic string $\varphi (x)$ subject to a Gaussian random potential $V(\varphi ,x)$ 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 $U(\varphi )$ and introduce two types of approximations amenable to exact solutions. Expanding the disorder potential $V(\varphi ,x)\approx V_0(x)+f(x)\varphi (x)$ 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 $U(\varphi )$ in a consistent manner. Using direct averaging as well as the replica technique, we derive the distribution functions ${\mathcal{P}}_{L,y}(F)$ and ${\mathcal{P}}_L(F)$ of free energies $F$ of a polymer of length $L$ for both fixed ($\varphi (L)=y$) and free boundary conditions on the displacement field $\varphi (x)$ 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 $U(\varphi )$.

16 pages, 5 figures

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