On the Layering Transition of an SOS Surface

We continue our study of the statistical mechanics of a 2D surface above a xed wall and attracted towards it by means of a very weak positive magnetic eld h in the solid on solid (SOS) approximation, when the inverse temperature is very large. In particular we consider a Glauber dynamics for the above model and study the rate of approach to equilibrium in a large cube with arbitrary boundary conditions. Using the results proved in the rst paper of this series we show that for all h 2 (h k+1 ; h k) (fh k g being the critical values of the magnetic eld found in the previous paper) the gap in the spectrum of the generator of the dynamics is bounded away from zero uniformly in the size of the box and in the boundary conditions. On the contrary, for h = h k and free boundary conditions, we show that the gap in a cube of side L is bounded from above and from below by a negative exponential of L. Our results provide a strong indication that, contrarily to what happens in two dimensions, for the three dimensional dynamical Ising model in a nite cube at low temperature and very small positive external eld, with boundary conditions that are opposite to the eld on one face of the cube and are absent (free) on the remaining faces, the rate of exponential convergence to equilibrium, which is positive in innnite volume, may go to zero exponentially fast in the side of the cube. This paper is the second part of a work, begun in CM], about the equilibrium and non equilibrium statistical mechanics of a SOS surface above a xed wall at low temperature and attracted towards it by a very weak external eld. The equilibrium distribution of the model in a nite volume V Z 2 with boundary conditions f(y)g y2Z 2 nV is described by the following Gibbs measure: