Dobrushin-Kotecky-Shlosman theorem for polygonal Markov fields in the plane

Details Dobrushin-Kotecky-Shlosman theorem for polygonal Markov fields in the plane Dobrushin-Kotecky-Shlosman theorem for polygonal Markov fields in the plane

2004-11-18

arxiv.org/abs/math-ph/0411064v2

Journal of Statistical Physics, {\bf 123}, pp. 631-684 (2006)

Physics

Mathematical Physics

59 pages, new version revised according to the referee's suggestions and now published

Scientific paper

10.1007/s10955-006-9053-7

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-Kotecky-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 Fernandez, Ferrari and Garcia.

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