  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 $U(\phi)$.
