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Powstawanie kropli dla wielokątnych pól Markowa

Schreiber Tomasz

Wiadomości Matematyczne

2008

T. 44, Nr 1

15--22

The paper discusses a class of continuum random contour models in the plane, going under the name of polygonal Markov fields as originally introduced by Arak & Surgailis and sharing a number of crucial features with the two-dimensional Ising model, which makes them interesting from the viewpoint of mathematical statistical physics. For such systems, modeling the co-existence of two opposing phases separated by polygonal contours, we present our results on low-temperature geometry of phase-separating interfaces. In this context, we show that in the phase transition regime the surplus of dominated phase creates a disk-shaped droplet surrounded by ocean of dominating phase (Wulff body) and minimising the model-specific surface energy functional. The proof is based on a particular graphical construction which also found its applications in digital image segmentation as indicated at the end of this article.