  Recently, it was shown that the probability distribution function (PDF) of
the free energy of a single continuum directed polymer (DP) in a random
potential, equivalently of the height of a growing interface described by the
Kardar-Parisi-Zhang (KPZ) equation, converges at large scale to the Tracy-Widom
distribution.The latter describes the fluctuations of the largest eigenvalue of
a random matrice, drawn from the Gaussian Unitary Ensemble (GUE), and the
result holds for a DP with fixed endpoints, i.e. for the KPZ equation with
droplet initial conditions. A more general conjecture can be put forward,
relating the free energies of $N>1$ non-crossing continuum DP in a random
potential, to the $N$-th largest eigenvalues of the GUE. Here, using replica
methods, we provide an important test of this conjecture by calculating exactly
the right tails of both PDF's and showing that they coincide for arbitrary $N$.
