Random dynamics and thermodynamic limits for polygonal Markov fields in the plane

We construct random dynamics on collections of non-intersecting planar contours, leaving invariant the distributions of lengthand area-interacting polygonal Markov fields with V-shaped nodes. The first of these dynamics is based on the dynamic construction of consistent polygonal fields, as presented in the original articles by Arak (1982) and Arak & Surgailis (1989, 1991), and it provides an easy-to-implement Metropolis-type simulation algorithm. The second dynamics leads to a graphical construction in the spirit of Fernández, Ferrari & Garcia (1998,2002) and it yields a perfect simulation scheme in a finite window from the infinite-volume limit. This algorithm seems difficult to implement, yet its value lies in that it allows for theoretical analysis of thermodynamic limit behaviour of length-interacting polygonal fields. The results thus obtained include the uniqueness and exponential α-mixing of the thermodynamic limit of such fields in the low temperature region, in the class of infinite-volume Gibbs measures without infinite contours. Outside this class we conjecture the existence of an infinite number of extreme phases breaking both the translational and rotational symmetries.