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 temperatureT _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 parameterb, related to the contact interaction, is positive,T _w is independent ofb and there is a logarithmic specific heat divergence asT _w is approached fromeither side. Here we show that forb<0,T _w does depend onb and there isno thermodynamic singularity from the wet phase. The partially wet phases forb0 andb>0 differ in the absence or presence of a monolayer covering the entire substrate; this results in a first-order transition across the lineb=0,T<T _w.This paper is dedicated to Jerry Percus.