A Polyhedral Markov Field - Pushing the Arak-Surgailis Construction into Three Dimensions

#### T. Schreiber

2006, v.12, Issue 1, 43-58

ABSTRACT

The purpose of the paper is to construct a polyhedral Markov field in $R^3$ in analogy with the planar construction of the original Arak polygonal Markov field [On Markovian random fields with finite number of values. 4th USSR-Japan Symposium on Probability Theory and Mathematical Statistics. Abstracts of Communications, Tbilisi, 1982]. We provide a dynamic construction of the process in terms of evolution of two-dimensional multi-edge systems tracing polyhedral boundaries of the field in three-dimensional time-space. We also give a general algorithm for simulating Gibbsian modifications of the constructed polyhedral field.

Keywords: Arak-Surgailis polygonal fields,polyhedral Markov fields,Gibbs measures,interacting particle systems