Free-energy distribution functions for the randomly forced directed polymer

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)≈V₀(x)+f(x)ϕ(x) at short distances, we study the random-force (or Larkin) problem with V₀(x)=0 as well as the shifted random-force problem including the random offset V₀(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 P_L,y(F) and 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 nonspectral correlator; we discuss how to implement this approximation in a proper way and present a general criterion for physically admissible disorder correlators U(ϕ) .