Abstract
We prove that for finite range discrete spin systems on the two dimensional latticeZ 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 ofZ 2, including the full lattice.
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Communicated by J.L. Lebowitz
Work partially supported by grant SC1-CT91-0695 of the Commission of European Communities and by grant DMS 91-00725 of the American NSF.
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Martinelli, F., Olivieri, E. & Schonmann, R.H. For 2-D lattice spin systems weak mixing implies strong mixing. Commun.Math. Phys. 165, 33–47 (1994). https://doi.org/10.1007/BF02099735
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DOI: https://doi.org/10.1007/BF02099735