On the Layering Transition of an SOS Surface Interacting with a Wall. I. Equilibrium Results

We consider the model of a 2D surface above a fixed wall and attracted toward it by means of a positive magnetic field h in the solid-on-solid (SOS) approximation when the inverse temperature beta is very large and the external field h is exponentially small in beta. We improve considerably previous results by Dinaburg and Mazel on the competition between the external field and the entropic repulsion with the wall, leading, in this case, to the phenomenon of layering phase transitions. In particular, we show, using the Pirogov-Sinai scheme as given by Zahradnik, that there exists a unique critical value h*(k)(beta) in the interval (1/4e(-4 beta k), 4e(-4 beta k)) such that, for all h is an element of (h*(k+1), h*(k)) and beta large enough, there exists a unique infinite-volume Gibbs state. The typical configurations are small perturbations of the ground state represented by a surface at height k+1 above the wall. Moreover, for the same choice of the thermodynamic parameters, the influence of the boundary conditions of the Gibbs measure in a finite cube decays exponentially fast with the distance from the boundary. When h=h*(k)(beta) we prove instead the convergence of the cluster expansion for both k and k+1 boundary conditions. This fact signals the presence of a phase transition. In the second paper of this series we will consider a Glauber dynamics for the above model and we will study the rate of approach to equilibrium in a large finite cube with arbitrary boundary conditions as a function of the external field ii. Using the results proven in this paper, we will show that there is a dramatic slowing down in the approach to equilibrium when the magnetic field takes one of the critical values and the boundary conditions are free (absent).