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 1990 |
ASJC Scopus subject areas
- Analysis
- Statistics and Probability
- Statistics, Probability and Uncertainty