  We consider the random wetting transition on the Cayley tree, i.e. the
problem of a directed polymer on the Cayley tree in the presence of random
energies along the left-most bonds. In the pure case, there exists a
first-order transition between a localized phase and a delocalized phase, with
a correlation length exponent $\nu_{pure}=1$. In the disordered case, we find
that the transition remains first-order, but that there exists two diverging
length scales in the critical region : the typical correlation length diverges
with the exponent $\nu_{typ}=1$, whereas the averaged correlation length
diverges with the bigger exponent $\nu_{av}=2$ and governs the finite-size
scaling properties. We describe the relations with previously studied models
that are governed by the same "Infinite Disorder Fixed Point". For the present
model, where the order parameter is the contact density $\theta_L=l_a/L$
(defined as the ratio of the number $l_a$ of contacts over the total length
$L$), the notion of "infinite disorder fixed point" means that the thermal
fluctuations of $\theta_L$ within a given sample, become negligeable at large
scale with respect to sample-to-sample fluctuations. We characterize the
statistics over the samples of the free-energy and of the contact density. In
particular, exactly at criticality, we obtain that the contact density is not
self-averaging but remains distributed over the samples in the thermodynamic
limit, with the distribution ${\cal P}_{T_c}(\theta) = 1/(\pi \sqrt{\theta
(1-\theta)})$.
