Consistent Markov Edge Processes and Random Graphs

We discuss Markov edge processes $\{Y_e; e \in E\}$ defined on edges of a directed acyclic graph $(V, E)$ with the consistency property: $$ {\mathrm P}_{E'}(Y_e; e \in E') = {\mathrm P}_E(Y_e; e \in E') $$ for a large class of subgraphs $(V',E')$ of $(V,E)$ obtained through a mesh dismantling algorithm. The probability distribution ${\mathrm P}_E$ of such edge process is a discrete version of consistent polygonal Markov graphs studied in \cite{arakDS1993, arakDS1989}. The class of Markov edge processes is related to the class of Bayesian networks and may be of interest to causal inference and decision theory. On regular $\nu$-dimensional lattices, consistent Markov edge processes have similar properties to Pickard random fields on ${\mathbb Z}^2$, representing a far-reaching extension of the latter class. A particular case of binary consistent edge process on ${\mathbb Z}^3$ was disclosed by Arak in a private communication. We prove that symmetric binary Pickard model generates Arak model on ${\mathbb Z}^2$ as a contour model.