Markov fields on branching planes

Charles Newman, C. Chris Wu

Research output: Contribution to journalArticle

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 languageEnglish (US)
Pages (from-to)539-552
Number of pages14
JournalProbability Theory and Related Fields
Volume85
Issue number4
DOIs
StatePublished - Dec 1990

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Gibbs Distribution
Branching
Free Boundary
Ising
Ising Model
Boundary conditions
Dependent
Line
Model

ASJC Scopus subject areas

  • Statistics and Probability
  • Analysis
  • Mathematics(all)

Cite this

Markov fields on branching planes. / Newman, Charles; Wu, C. Chris.

In: Probability Theory and Related Fields, Vol. 85, No. 4, 12.1990, p. 539-552.

Research output: Contribution to journalArticle

Newman, Charles ; Wu, C. Chris. / Markov fields on branching planes. In: Probability Theory and Related Fields. 1990 ; Vol. 85, No. 4. pp. 539-552.
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