Consider a binary Markov random field whose neighbor structure is specified by a countable graph with nodes of uniformly bounded degree. Under a minimal assumption we prove a decomposition theorem to the effect that such a Markov random field can be represented as the nodewise modulo 2 sum of two independent binary random fields, one of which is white binary noise of positive weight. Said decomposition provides the information theorist with an exact expression for the per-site rate-distortion function of the random field over an interval of distortions not exceeding this weight. We mention possible implications for communication theory, probability theory and statistical physics.
"A Decomposition Theorem for Binary Markov Random Fields." Ann. Probab. 15 (3) 1112 - 1125, July, 1987. https://doi.org/10.1214/aop/1176992084