  The statistical properties of a two dimensional lattice of elastic lines in a
random medium are studied using the Bethe ansatz. We present a novel mapping of
the dilute random line lattice onto the weak coupling limit of a pure Bose gas
with delta-function interactions. Using this mapping, we calculate the
cumulants of the free energy in the dilute limit exactly. The relation between
density and displacement correlation functions in the two models is examined
and compared with existing results from renormalization group and variational
ansaetze.
