The wetting transition in a random surface model

We continue our analysis of the phase diagram of a discrete random surface, with no "downward fingers," lying above a flat two-dimensional substrate. The surface is closely related to the 2D Ising model and its free energy is exactly solvable in much (but not all) of the phase diagram. There is a transition at temperature T_w from a high-T infinite height or wet phase to a low-T finite height or partially wet phase. Previously it was shown that when a parameter b, related to the contact interaction, is positive, T_w is independent of b and there is a logarithmic specific heat divergence as T_w is approached from either side. Here we show that for b<0, T_w does depend on b and there is no thermodynamic singularity from the wet phase. The partially wet phases for b≤0 and b>0 differ in the absence or presence of a monolayer covering the entire substrate; this results in a first-order transition across the line b=0, T<T_w.