Joint free-energy distribution in the random directed polymer problem

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 Fover ] and the free-energy difference F;{'} of the two configurations, we determine the joint distribution function P(L,u)(Fover ],F(')) using the replica approach. We find that for large L and large negative free energies Fover ], the joint distribution function factorizes into longitudinal P(L,u)(Fover ])] and transverse P(u)(F('))] components, which furthermore coincide with results obtained previously via different independent routes.