### Abstract

We investigate Ising models indexed by the sites of a branching plane {Mathematical expression} × ℤ, which is the product of a regular tree {Mathematical expression} and the lineℤ. There are three parameter regimes corresponding to: (1) a unique Gibbs distribution; (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) 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 {Mathematical expression} × ℤ, the Fortuin-Kasteleyn representation of Ising (and Potts) systems as dependent percolation models, and a "finite island" property of percolation models on {Mathematical expression} × ℤ.

Original language | English (US) |
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Pages (from-to) | 539-552 |

Number of pages | 14 |

Journal | Probability Theory and Related Fields |

Volume | 85 |

Issue number | 4 |

DOIs | |

State | Published - Dec 1 1990 |

### All Science Journal Classification (ASJC) codes

- Statistics and Probability
- Analysis
- Mathematics(all)

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## Cite this

*Probability Theory and Related Fields*,

*85*(4), 539-552. https://doi.org/10.1007/BF01203170