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Non-homogeneous Polygonal Markov Fields in the Plane: Graphical Representations and Geometry of Higher Order Correlations

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Abstract

We consider polygonal Markov fields originally introduced by Arak and Surgailis (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 features with the two-dimensional Ising model. We introduce non-homogeneous version of polygonal fields in anisotropic environment. For these fields we provide a class of new graphical constructions and random dynamics. These include a generalized dynamic representation, generalized and defective disagreement loop dynamics as well as a generalized contour birth and death dynamics. Next, we use these constructions as tools to obtain new exact results on the geometry of higher order correlations of polygonal Markov fields in their consistent regime.

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Authors and Affiliations

  1. Faculty of Mathematics and Computer Science, Nicolaus Copernicus University, ul. Chopina 12/18, 87-100, Toruń, Poland

    Tomasz Schreiber

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  1. Tomasz Schreiber
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Correspondence to Tomasz Schreiber.

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Research supported by the Polish Minister of Science and Higher Education grant N N201 385234 (2008-2010).

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Schreiber, T. Non-homogeneous Polygonal Markov Fields in the Plane: Graphical Representations and Geometry of Higher Order Correlations. J Stat Phys 132, 669–705 (2008). https://doi.org/10.1007/s10955-008-9584-1

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  • DOI: https://doi.org/10.1007/s10955-008-9584-1

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