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Abstract.We consider the metastable behavior in the so-called pathwise approach of a ferromagnetic spin system with a Glauber dynamics in a finite two dimensional torus under a positive magnetic field in the limit as the temperature goes to zero. First we consider the evolution starting from a single rectangular droplet of spins + 1 in a sea of spins -1. We show that small droplets are likely to disappear while large droplets are likely to grow; the threshold between the two cases being sharply defined and depending only on the external field. This result is used to prove that starting from the configuration with all spins down (-1) the pattern of evolution leading to the more stable configuration with all spins up (+ 1_) approaches, as the temperature vanishes, a metastable behavior: the system stays close to -1 for an unpredictable time until a critical square droplet of a precise size is eventually formed and nucleates the decay to +_1 in a relatively short time. The asymptotic magnitude of the total decay time is shown to be related to the height of an energy barrier, as expected from heuristic and mean field studies of metastability.

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We prove that for finite range discrete spin systems on the two dimensional lattice Z 2 , the (weak) mixing condition which follows, for instance, from the Dobrushin-Shlosman uniqueness condition for the Gibbs state implies a stronger mixing property of the Gibbs state, similar to the Dobrushin-Shlosman complete analyticity condition, but restricted to all squares in the lattice, or, more generally, to all sets multiple of a large enough square. The key observation leading to the proof is that a change in the boundary conditions cannot propagate either in the bulk, because of the weak mixing condition, or along the boundary because it is one dimensional. As a consequence we obtain for ferromagnetic Ising-type systems proofs that several nice properties hold arbitrarily close to the critical temperature; these properties include the existence of a convergent cluster expansion and uniform boundedness of the logarithmic Sobolev constant and rapid convergence to equilibrium of the associated Glauber dynamics on nice subsets of Z 2 , including the full lattice.

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