 93154 F.Martinelli , E.Olivieri , R.H.Schonmann
 For 2D lattice spin systems Weak Mixing implies Strong Mixing
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May 28, 93

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Abstract. We prove that for
finite range discrete spin systems on the two dimensional lattice
the (weak) mixing condition which follows, for instance, from the
DobrushinShlosman uniqueness condition for the Gibbs
state implies a stronger mixing property of the Gibbs state,
similar to the DobrushinShlosman complete analiticity 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 Isingtype 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 the lattice, including the full lattice.
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