FOR 2-D LATTICE SPIN SYSTEMS WEAK MIXING IMPLIES STRONG MIXING

We prove that for finite range discrete spin systems on the two dimens ional lattice Z2, the (weak) mixing condition which follows, for insta nce, from the Dobrushin-Shlosman uniqueness condition for the Gibbs st ate 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 multipl e of a large enough square. The key observation leading to the proof i s 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 ferromag netic Ising-type systems proofs that several nice properties hold arbi trarily close to the critical temperature; these properties include th e existence of a convergent cluster expansion and uniform boundedness of the logarithmic Sobolev constant and rapid convergence to equilibri um of the associated Glauber dynamics on nice subsets of Z2, including the full lattice.