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.