Communications in Mathematical Physics

, Volume 165, Issue 1, pp 33–47 | Cite as

For 2-D lattice spin systems weak mixing implies strong mixing

  • F. Martinelli
  • E. Olivieri
  • R. H. Schonmann


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.


Critical Temperature Spin System System Proof Nice Property Lattice Spin 
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Copyright information

© Springer-Verlag 1994

Authors and Affiliations

  • F. Martinelli
    • 1
  • E. Olivieri
    • 2
  • R. H. Schonmann
    • 3
  1. 1.Dipartimento di MatematicaIII Università di RomaItaly
  2. 2.Dipartimento di MatematicaUniversità “Tor Vergata”RomaItaly
  3. 3.Mathematics DepartmentUCLALos AngelesUSA

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