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arXiv · 1007.0852

Free-energy distribution functions for the randomly forced directed polymer

V. S. Dotsenko·V. B. Geshkenbein·D. A. Gorokhov·G. Blatter
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Abstract

We study the $1+1$-dimensional random directed polymer problem, i.e., an elastic string $ϕ(x)$ subject to a Gaussian random potential $V(ϕ,x)$ 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 $U(ϕ)$ and introduce two types of approximations amenable to exact solutions. Expanding the disorder potential $V(ϕ,x) \approx V_0(x) + f(x) ϕ(x)$ 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 $U(ϕ)$ 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 ($ϕ(L) = y$) and free boundary conditions on the displacement field $ϕ(x)$ 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 $U(ϕ)$.

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V. S. Dotsenko, V. B. Geshkenbein, D. A. Gorokhov, G. Blatter. 2010-07-06. Free-energy distribution functions for the randomly forced directed polymer. https://doi.org/10.1103/physrevb.82.174201

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