Multi-colour random fields with polygonal realisations

We introduce a class of random fields that can be understood as discrete versions of multi-colour polygonal fields built on regular linear tessellations. We focus first on consistent polygonal fields, for which we show consistency, Markovianity, and solvability by means of a dynamic representation. This representation also forms the basis for new sampling techniques for Gibbsian modifications of such fields, a class which covers lattice based random fields.