On matrix technique for binary Markov fields on lattices

We consider a plane rectangular lattice formed by the intersection of L verticals with M horizontals of lengths M+1 and L+1 respectively. Thus the lattice consists of LM nodes and 2LM+L+M branches. The branches are assumed to be binary variables interacting through the lattice nodes where a (0,1)-interaction matrix B=(b_i2,j2), b∈{0,1}, i ²,j²∈{0,1}² is placed. Let Φ(L,M,B) be a binary Markov LxM-field generated by B. We are interested in finding the number of all field realizations |Φ(L,M,B)| and presenting it as a function of the field parameters L,M,B. In the paper we develop a matrix technique similar to that which governs the discrete Markov chains. The matrix technique for two- (and more) dimensional lattices depends on the topology of the lattice; therefore the main feature of the problem is finding an adequate matrix description of the lattice topology. We analyze the problem in two modifications: in the plane modification described above and in a described cylindrical modification