Markov fields on branching planes

Charles M. Newman, C. Chris Wu

Research output: Contribution to journalArticlepeer-review


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 languageEnglish (US)
Pages (from-to)539-552
Number of pages14
JournalProbability Theory and Related Fields
Issue number4
StatePublished - Dec 1990

ASJC Scopus subject areas

  • Analysis
  • Statistics and Probability
  • Statistics, Probability and Uncertainty


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