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

Abstract

We construct random dynamics for collections of nonintersecting 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 (1983) 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 Fernández et al. (1998), (2002) and yields a perfect simulation scheme in a finite window in the infinite-volume limit. This algorithm seems difficult to implement, yet its value lies in that it allows for theoretical analysis of the thermodynamic limit behaviour of length-interacting polygonal fields. The results thus obtained include, in the class of infinite-volume Gibbs measures without infinite contours, the uniqueness and exponential α-mixing of the thermodynamic limit of such fields in the low-temperature region. Outside this class, we conjecture the existence of an infinite number of extreme phases breaking both the translational and rotational symmetries.

Non-homogeneous Polygonal Markov Fields in the Plane: Graphical Representations and Geometry of Higher Order Correlations

“…Consequently, we conclude from Lemma 5 that under Θ [β] , β ≥ 2, the contour size exhibits exponentially decaying tails, which is a non-homogeneous counterpart of Lemma 1 in [15].…”

“…These observations place us within the framework of the general contour birth and death graphical construction as developed by Fernández, Ferrari & Garcia [7,8,9] and as sketched below, see ibidem and [15] for further details. Choose β large enough, to be specified below.…”

“…We also provide a modification of this dynamics which can be used for Monte-Carlo simulation of A M;β D for all β ∈ R. We build upon [15] in our presentation of the standard dynamics based on an important concept of a disagreement loop.…”

“…Under even milder conditions one can guarantee the isometry invariance of the field as well as the two-dimensional germ-Markov property stating that the conditional behaviour of the field in an open bounded domain depends on the exterior configuration only through arbitrarily close neighbourhoods of the boundary, see ibidem. A particularly interesting class of processes seem to be length-and area-interacting modifications of consistent fields, which not unexpectedly exhibit many features analogous to those of the two-dimensional Ising model, including the presence of first order phase transition at low enough temperatures (Nicholls,[14]; Schreiber, [15]) and low-temperature phase separation phenomenon (Schreiber,[16]). Consistent polygonal fields and their length-interacting modifications have also interesting statistical applications where they are used as priors in Bayesian image analysis (Clifford & Nicholls,[6]; Kluszczyński, van Lieshout & Schreiber, [10,11]; van Lieshout & Schreiber, [12,17]).…”

“…These were first provided by Arak & Surgailis [2,3] and Arak, Clifford & Surgailis [4] in the form of the so-called dynamic representations for consistent polygonal fields. Later, we have introduced disagreement loop dynamics and contour birth and death representation for broader classes of polygonal fields, see [15]. The purpose of the present paper is twofold.…”