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

摘要

We construct random dynamics on collections of non-intersecting planarcontours, leaving invariant the distributions of length- and area-interactingpolygonal Markov fields with V-shaped nodes. The first of these dynamics isbased on the dynamic construction of consistent polygonal fields, as presentedin the original articles by Arak (1982) and Arak and Surgailis (1989, 1991),and it provides an easy-to-implement Metropolis-type simulation algorithm. Thesecond dynamics leads to a graphical construction in the spirit of Fernandez,Ferrari and Garcia (1998,2002) and it yields a perfect simulation scheme in afinite window from the infinite-volume limit. This algorithm seems difficult toimplement, yet its value lies in that it allows for theoretical analysis ofthermodynamic limit behaviour of length-interacting polygonal fields. Theresults thus obtained include the uniqueness and exponential $\alpha$-mixing ofthe thermodynamic limit of such fields in the low temperature region, in theclass of infinite-volume Gibbs measures without infinite contours. Outside thisclass we conjecture the existence of an infinite number of extreme phasesbreaking both the translational and rotational symmetries