Random partial orders, posts, and the causal set approach to discrete quantum gravity

We study a collection of Markov chains with values in the collection of partial orderings of the natural numbers. These systems arise naturally in the causal set approach to discrete quantum gravity and include the well-studied random partial orders of Alon, Bollobas, Brightwell, and Janson [Ann. Appl. Probab. 4, 108-123 (1994)]. We prove that under the dynamics associated to Markov chains in our collection, posts occur infinitely often, almost surely. (C) 2005 American Institute of Physics.