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

Abstract

We construct random dynamics on collections of non-intersecting planar contours, leaving invariant the distributions of length- and 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 and 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 Fernandez, Ferrari and 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