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We consider polygonal Markov fields originally introduced by Arak andSurgailis (1989). Our attention is focused on fields with nodes of order two,which can be regarded as continuum ensembles of non-intersecting contours inthe plane, sharing a number of features with the two-dimensional Ising model.We introduce non-homogeneous version of polygonal fields in anisotropicenviroment. For these fields we provide a class of new graphical constructionsand random dynamics. These include a generalised dynamic representation,generalised and defective disagreement loop dynamics as well as a generalisedcontour birth and death dynamics. Next, we use these constructions as tools toobtain new exact results on the geometry of higher order correlations ofpolygonal Markov fields in their consistent regime.

Non-homogeneous polygonal Markov fields in the plane: graphical representations and geometry of higher order correlations