APPROACH TO EQUILIBRIUM OF GLAUBER DYNAMICS IN THE ONE-PHASE REGION .1. THE ATTRACTIVE CASE

Various finite volume mixing conditions in classical statistical mecha nics are reviewed and critically analyzed. In particular some finite s ize conditions are discussed, together with their implications for the Gibbs measures and for the approach to equilibrium of Glauber dynamic s in arbitrarily large volumes. It is shown that Dobrushin-Shlosman's theory of complete analyticity and its dynamical counterpart due to St roock and Zegarlinski, cannot be applied, in general, to the whole one phase region since it requires mixing properties for regions of arbit rary shape. An alternative approach, based on previous ideas of Olivie ri, and Picco, is developed, which allows to establish results on rapi d approach to equilibrium deeply inside the one phase region. In parti cular, in the ferromagnetic case, we considerably improve some previou s results by Holley and Aizenman and Holley. Our results are optimal i n the sense that, for example, they show for the first time fast conve rgence of the dynamics for any temperature above the critical one for the d-dimensional Ising model with or without an external field. In pa rt II we extensively consider the general case (not necessarily attrac tive) and we develop a new method, based on renormalizations group ide as and on an assumption of strong mixing in a finite cube, to prove hy percontractivity of the Markov semigroup of the Glauber dynamics.