Random wetting transition on the Cayley tree : a disordered first-order transition with two correlation length exponents

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 ${\cal P}_{T_c}(\theta) = 1/(\pi \sqrt{\theta (1-\theta)})$.