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Markov fields on branching planes
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  • Published: December 1990

Markov fields on branching planes

  • Charles M. Newman1 nAff2 &
  • C. Chris Wu1 nAff2 

Probability Theory and Related Fields volume 85, pages 539–552 (1990)Cite this article

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Summary

We investigate Ising models indexed by the sites of a branching plane\(\mathbb{T}\) × ℤ, which is the product of a regular tree\(\mathbb{T}\) and the lineℤ. There are three parameter regimes corresponding to:

  1. (1)

    a unique Gibbs distribution;

  2. (2)

    nonunique Gibbs distributions with treelike structure — the free boundary condition field is not a mixture of the plus and minus b.c. fields;

  3. (3)

    nonunique Gibbs distributions with planelike structure — the free b.c. field is a mixture of the plus and minus b.c. fields.

Our analysis is based on earlier work by Grimmett and Newman concerning independent percolation on\(\mathbb{T}\) × ℤ, the Fortuin-Kasteleyn representation of Ising (and Potts) systems as dependent percolation models, and a “finite island” property of percolation models on\(\mathbb{T}\) × ℤ.

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Author information

Author notes
  1. Charles M. Newman & C. Chris Wu

    Present address: Courant Institute of Mathematical Sciences, New York University, 10012, New York, NY, USA

Authors and Affiliations

  1. Department of Mathematics and Program in Applied Mathematics, University of Arizona, 85721, Tucson, AZ, USA

    Charles M. Newman & C. Chris Wu

Authors
  1. Charles M. Newman
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  2. C. Chris Wu
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Additional information

Research supported in part by NSF grants DMS-8514834 and DMS-8902156, by AFOSR grant 88-0189 and by AFOSR contract F 49620-86-C0130 to the Arizona Center for Mathematical Sciences

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Cite this article

Newman, C.M., Wu, C.C. Markov fields on branching planes. Probab. Th. Rel. Fields 85, 539–552 (1990). https://doi.org/10.1007/BF01203170

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  • Received: 10 August 1989

  • Revised: 10 January 1990

  • Issue Date: December 1990

  • DOI: https://doi.org/10.1007/BF01203170

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Keywords

  • Boundary Condition
  • Stochastic Process
  • Condition Field
  • Probability Theory
  • Free Boundary
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