  We consider two configurations of a random directed polymer of length L
confined to a plane and ending in two points separated by 2u. Defining the mean
free energy $\bar F$ and the free energy difference F' of the two
configurations, we determine the joint distribution function P_{L,u}(\bar F,F')
using the replica approach. We find that for large L and large negative free
energies $\bar F$, the joint distribution function factorizes into longitudinal
(P_{L,u} (\bar F)) and transverse (P_{u}(F')) components, which furthermore
coincide with results obtained prevously via different independent routes.
