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

TL;DR

This paper proves a phase separation theorem for polygonal Markov fields in the plane, showing a disk-shaped Wulff shape at low temperatures, using a graphical contour process instead of traditional cluster expansions.

Contribution

It establishes a Dobrushin-Kotecky-Shlosman type phase separation result for V-shaped polygonal Markov fields, introducing a novel graphical construction method.

Findings

Proves low-temperature phase separation for polygonal fields.

Identifies the Wulff shape as a disk due to rotational invariance.

Develops a graphical contour process as a new analytical tool.

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-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.