Non-typical Wulff shapes in a corner: A microscopic derivation

A complete microscopic analysis of the equilibrium shape of a droplet in a corner between two walls is given within a Gaussian SOS model. We derive a statistical mechanical proof of the Winterbottom and the Summertop constructions for the equilibrium shapes, including a proof of generalized Young relations for inclined walls. We discuss a phase diagram with convexity-concavity transitions and wetting transitions induced by changing the inclination of the walls. A possible degeneracy of the solutions of the thermodynamic variational problem at the convexity-concavity transition point is discussed in the Gaussian model from a statistical mechanical point of view.