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Journal of Statistical Physics

, Volume 123, Issue 3, pp 631–684 | Cite as

Dobrushin-Kotecký-Shlosman Theorem for Polygonal Markov Fields in the Plane

  • Tomasz Schreiber
Article

Abstract

We consider the so-called length-interacting Arak-Surgailis polygonal Markov fields with V-shaped nodes — a continuum and isometry invariant process in the plane sharing a number of properties with the two-dimensional Ising model. For these polygonal fields we establish a low-temperature phase separation theorem in the spirit of the Dobrushin-Kotecký-Shlosman theory, with the corresponding Wulff shape deteremined to be a disk due to the rotation invariant nature of the considered model. As an important tool replacing the classical cluster expansion techniques and very well suited for our geometric setting we use a graphical construction built on contour birth and death process, following the ideas of Férnandez, Ferrari and Garcia.

Keywords

phase separation DKS theorem Wulff shape Arak-Surgailis polygonal Markov fields 

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Copyright information

© Springer Science+Business Media, Inc. 2006

Authors and Affiliations

  1. 1.Faculty of Mathematics & Computer ScienceNicolaus Copernicus UniversityToruńPoland

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