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

TL;DR

This paper introduces an explicit stochastic representation for higher-order correlation functions of polygonal Markov fields in the plane, using polygonal webs and crop functionals, linking graphical constructions with probabilistic analysis.

Contribution

It provides a novel explicit formula for correlation functions of polygonal Markov fields via polygonal webs, extending understanding of their structure and properties.

Findings

Derived an explicit stochastic representation for correlation functions

Established a martingale interpolation connecting fields and webs

Enhanced analytical tools for polygonal Markov fields

Abstract

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.