Polygonal Web Representation for Higher Order Correlation Functions of Consistent Polygonal Markov Fields in the Plane

We consider polygonal Markov fields originally introduced by Arak in 4th USSR-Japan Symposium on Probability Theory and Mathematical Statistics, Abstracts of Communications, 1982; Arak and Surgailis in Probab. Theory Relat. Fields 80:543-579, 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 Mobius-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.