Surface tension and SOS limit in the 2D Ising model

The Sherman theorem on closed paths for the two-dimensional Ising model on a square with + boundary conditions is generalised to arbitrary boundary conditions. The refinement allows a rigorous cluster expansion for boundary observables (in particular for the surface tension) in terms of open random trajectories on the lattice. As an application the authors discuss the grand canonical cluster expansion for the surface tension and prove its convergence to the canonical cluster expansion and to the SOS limit.