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We consider polygonal Markov fields originally introduced by Arak andSurgailis (1982,1989).  Our attention is focused on fields with nodes of order two, which can beregarded as continuum ensembles of non-intersecting contours in the plane,sharing a number of salient features with the two-dimensional Ising model. Thepurpose of this paper is to establish an explicit stochastic representation forthe higher-order correlation functions of polygonal Markov fields in theirconsistency regime. The representation is given in terms of the so-called cropfunctionals (defined by a Moebius-type formula) of polygonal webs which arisein a graphical construction dual to that giving rise to polygonal fields. Theproof of our representation formula goes by constructing a martingaleinterpolation between the correlation functions of polygonal fields and cropfunctionals of polygonal webs.

Polygonal web representation for higher order correlation functions of consistent polygonal Markov fields in the plane