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We consider two configurations of a random directed polymer of length Lconfined to a plane and ending in two points separated by 2u. Defining the meanfree energy $\bar F$ and the free energy difference F' of the twoconfigurations, 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 freeenergies $\bar F$, the joint distribution function factorizes into longitudinal(P_{L,u} (\bar F)) and transverse (P_{u}(F')) components, which furthermorecoincide with results obtained prevously via different independent routes.

Joint free energy distribution in the random directed polymer problem