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On weak mixing in lattice models
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  • Published: May 1998

On weak mixing in lattice models

  • Kenneth S. Alexander1 

Probability Theory and Related Fields volume 110, pages 441–471 (1998)Cite this article

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Summary.

For lattice models on ℤd, weak mixing is the property that the influence of the boundary condition on a finite decays exponentially with distance from that region. For a wide class of models on ℤ2, including all finite range models, we show that weak mixing is a consequence of Gibbs uniqueness, exponential decay of an appropriate form of connectivity, and a natural coupling property. In particular, on ℤ2, the Fortuin-Kasteleyn random cluster model is weak mixing whenever uniqueness holds and the connectivity decays exponentially, and the q-state Potts model above the critical temperature is weak mixing whenever correlations decay exponentially, a hypothesis satisfied if q is sufficiently large. Ratio weak mixing is the property that uniformly over events A and B occurring on subsets Λ and Γ, respectively, of the lattice, |P(A∩B)/P(A)P(B)−1| decreases exponentially in the distance between Λ and Γ. We show that under mild hypotheses, for example finite range, weak mixing implies ratio weak mixing.

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Authors and Affiliations

  1. Department of Mathematics, DRB 155, University of Southern California, 1042 West 36th Place, Los Angeles, CA 90089-1113, USA, e-mail: alexandr@math.usc.edu, , , , , , US

    Kenneth S. Alexander

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  1. Kenneth S. Alexander
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Received: 27 August 1996 / In revised form: 15 August 1997

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Alexander, K. On weak mixing in lattice models. Probab Theory Relat Fields 110, 441–471 (1998). https://doi.org/10.1007/s004400050155

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  • Issue Date: May 1998

  • DOI: https://doi.org/10.1007/s004400050155

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  • Mathematics Subject Classification (1991): Primary 60K35
  • Secondary 85B20
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