Markov fields with polygonal realizations

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

“…The first one is the consistent regime corresponding to β = 1 and introduced in Section 3 -we shall argue that this particular choice of temperature parameter places us in the context of a non-homogeneous version of Arak-Surgailis [2] construction for the so-called consistent polygonal fields, thus ensuring the availability of an appropriate dynamic representation for our process in terms of equilibrium evolution of one-dimensional particle systems tracing the boundaries of the field in two-dimensional space-time, as discussed in detail in Subsection 3.1 below, see also [2]. The afore-mentioned The second regime in the focus of our interest is the low temperature region (large positive β) where long range point-to-point correlations are present, giving rise to the spontaneous magnetisation phenomenon, see Corollary 6 in Section 8 and the discussion below it.…”

“…The fields with nodes of order 2, or V-shaped nodes for short, which are in the focus of our attention in this paper, arise as ensembles of self-avoiding closed polygonal contours in the plane interacting by hard core exclusions, possibly with some further terms entering the Hamiltonian. Under a particular choice of the Hamiltonian the polygonal fields enjoy striking properties including consistency (the field constructed in a subdomain D ⊆ D ′ ⊆ R 2 coincides with the restriction to D of the field constructed in D ′ ) as well as availability of an explicit formula for the partition function and some other numerical characteristics of the field, see [2]. 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.…”

“…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].…”

“…Observe also that this construction should be regarded as a specific version of the general polygonal model given in (2.11) in [2]. The finiteness of the partition function…”

“…We shall also consider the empty boundary version of the above construction, with A M;β D|∅ arising in law as A M;β D conditioned on staying within Γ D|∅ , that is to say not containing boundary vertices. It is easily seen that alternatively A M;β D|∅ can be defined by rewriting (2) above with Γ D replaced with Γ D|∅ throughout. The corresponding partition function Z M;β D|∅ is always finite as well, see (31) below.…”