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)
a unique Gibbs distribution;
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(2)
nonunique Gibbs distributions with treelike structure — the free boundary condition field is not a mixture of the plus and minus b.c. fields;
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(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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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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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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DOI: https://doi.org/10.1007/BF01203170