  We consider polygonal Markov fields originally introduced by Arak and
Surgailis (1982,1989).
  Our attention is focused on fields with nodes of order two, which can be
regarded as continuum ensembles of non-intersecting contours in the plane,
sharing a number of salient features with the two-dimensional Ising model. The
purpose of this paper is to establish an explicit stochastic representation for
the higher-order correlation functions of polygonal Markov fields in their
consistency regime. The representation is given in terms of the so-called crop
functionals (defined by a Moebius-type formula) of polygonal webs which arise
in a graphical construction dual to that giving rise to polygonal fields. The
proof of our representation formula goes by constructing a martingale
interpolation between the correlation functions of polygonal fields and crop
functionals of polygonal webs.
