Duality relations and equilibrium crystal shapes of Potts models on triangular and honeycomb lattices

We consider the Q-state Potts models on triangular and honeycomb lattices. For each lattice it is shown that the anisotropic interfacial tension is related to the anisotropic correlation length on the dual lattice. Using the star-triangle transformation, we also find that the anisotropic correlation length on the triangular lattice is the same as that of the honeycomb lattice for Q>4 at the first-order transition point. The anisotropic correlation length is exactly calculated at the first-order transition point by a method which introduces auxiliary edges into the models. From the calculated anisotropic correlation length, the equilibrium droplet shape of one ordered phase embedded inside another is derived via the Wulff construction. The equilibrium shape is written as a simple algebraic curve, which corresponds to the dispersion relation of bound states (in the terminology of spin chains).