Sampling monotone Boolean functions
In [1], the paper where Propp &amp; Wilson introduce the "coupling from the past" technique for MCMC sampling, they also outline a "heat bath algorithm" applicable to a certain kind of spin system model (Section 3.1). Then, they further illustrate how to map their spin system model to the problem of uniformly sampling from the order ideals of a poset which, in turn, is equivalent to sampling from the antichains of that poset (Section 3.2). 


Propp, James Gary, and David Bruce Wilson. "Exact sampling with coupled Markov chains and applications to statistical mechanics." Random Structures &amp; Algorithms 9.1‐2 (1996): 223-252.