  We show that effective interactions mediated by disorder between two directed
polymers can be modelled as the crosscorrelation of noises in the
Kardar-Parisi-Zhang (KPZ) equations satisfied by the respective free energies
of these polymers. When there are two polymers, disorder introduces attractive
interactions between them. We analyze the phase diagram in details and show
that these interactions lead to new phases in the phase diagram. We show that,
even in dimension $d=1$, the two directed polymers see the attraction only if
the strength of the disorder potential exceeds a threshold value. We extend our
calculations to show that if there are $m$ polymers in the system then $m$-body
interactions are generated in the disorder averaged effective free energy.
