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