For 2-D lattice spin systems weak mixing implies strong mixing |
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Authors: | F. Martinelli E. Olivieri R. H. Schonmann |
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Affiliation: | (1) Dipartimento di Matematica, III Università di Roma, Italy;(2) Dipartimento di Matematica, Università Tor Vergata, Roma, Italy;(3) Mathematics Department, UCLA, 90024 Los Angeles, CA, USA |
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Abstract: | We prove that for finite range discrete spin systems on the two dimensional latticeZ2, 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 ofZ2, including the full lattice.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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