An Ising interface between two walls: Competition between two tendencies

We consider a ferromagnetic Ising spin system, consisting of m + 1, d-dimensional, layers with "-" boundary condition on the bottom layer and "+" on the top layer. When β is larger than β_cr, the inverse critical temperature for the d-dimensional Ising model, the interface generated by the boundary conditions is expected to be halfway between bottom and top, for m odd, and just above or below the middle layer, for m even (each possibility with probability 1/2). In this paper, we prove the above assertion under the condition that β ≥ const · m and partly for β > β_cr.