Interface Pinning and Finite Size Effects in the 2d Ising Model

We apply new techniques developed in [PV1] to the study of some surface effects in the 2D Ising model. We examine in particular the pinningdepinning transition. The results are valid for all subcritical temperatures. By duality we obtained new finite size effects on the asymptotic behaviour of the two– point correlation function above the critical temperature. The key–point of the analysis is to obtain good concentration properties of the measure defined on the random lines giving the high–temperature representation of the two–point correlation function, as a consequence of the sharp triangle inequality: let τ̂ (x) be the surface tension of an interface perpendicular to x; then for any x, y τ̂ (x) + τ̂ (y)− τ̂ (x+ y) ≥ 1 κ (‖x‖+ ‖y‖ − ‖x+ y‖) , where κ is the maximum curvature of the Wulff shape and ‖x‖ the Euclidean norm of x.