ON THE LAYERING TRANSITION OF AN SOS SURFACE INTERACTING WITH A WALL .2. THE GLAUBER DYNAMICS

We continue our study of the statistical mechanics of a 2D surface abo ve a fixed wall and attracted towards it by means of a very weak posit ive magnetic field h in the solid on solid (SOS) approximation, when t he inverse temperature beta is very large. In particular we consider a Glauber dynamics for the above model and study the rate of approach t o equilibrium in a large cube with arbitrary boundary conditions. Usin g the results proved in the first paper of this series we show that fo r all h is an element of (h(k+1), h(k)*) ({h(k)*} being the critical values of the magnetic field found in the previous paper) the gap in t he 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 t he 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 tha t, contrary to what happens in two dimensions, for the three dimension al dynamical Ising model in a finite cube al low temperature and very small positive external field, with boundary conditions that are oppos ite to the field on one face of the cube and are absent (free) on the remaining faces, the rate of exponential convergence to equilibrium, w hich is positive in infinite volume, may go to zero exponentially fast in the side of the cube.