Binary Markov mesh models and symmetric Markov random fields: Some results on their equivalence (Q5936985)

Consider a general two-dimensional Ising model whose (translation invariant) interactions correspond to subsets of the elementary square. If all such interactions are present, the Hamiltonian of the system becomes a function of 15 parameters. Even in the case of a symmetric Markov field (SMF), when the spatial dependence appears symmetrically in the conditional probabilities, the Gibbs measure still depends on 10 parameters, and a perfect sampling from the related Gibbs measure becomes quite a delicate task. Much better adapted for simulations are so-called partially ordered Markov models (POMM), where the dependence of a fixed spin from the ``past'' is realised through its (small) ``past'' neighbourhood. Due to existence of a simple closed representation of the Gibbs measure through the product of (local) conditional distributions for POMM the Gibbs sampler algorithm works much faster, and hence it is natural to ask which POMM give rise to SMF. The authors consider several subclasses of models from POMM (crystal-growth models (CGM), Markov mesh models (MMM), mutually compatible Markov random fields (MCMRF)), and in particular give a sufficient condition for an MMM to be an SMF. Because of a large amount of parameters it is not surprising that the corresponding result (Lemma 3.1) is almost three pages long! Also, the authors demonstrate usefulness of their results by analysing the efficiency of simulations on 64\(\times\)64 toroidal lattice.