Interface tension, equilibrium crystal shape, and imaginary zeros of partition function: Planar Ising systems.

Abstract

For any Ising model on a planar lattice without bond crossings, we prove that the anisotropic interface tension is given by the lattice Green's function of a free random-walk problem defined on the dual lattice. This fact, derived via Vdovichenko's combinatorial method, reveals a general relation between the bulk free energy and the interface tension for two-dimensional Ising models. As an important consequence, we show that the equilibrium crystal shape corresponds to imaginary zeros of the partition function. We also discuss generalizations to non-Ising and non-solvable systems.