We prove that bounded multiparameter martingales converge almost surely if the underlying $\sigma$-fields are generated by a Markov random field which satisfies Dobrushin's uniqueness condition. An example shows that it is not enough to assume that the Markov field is uniquely determined by its conditional probabilities.
"Almost Sure Convergence of Multiparameter Martingales for Markov Random Fields." Ann. Probab. 12 (1) 133 - 140, February, 1984. https://doi.org/10.1214/aop/1176993378