Markov chains for linear extensions, the two-dimensional case

We study the generation of uniformly distributed linear extensions using Markou chains. In particular we show that monotone coupling from the past can be applied in the case of linear extensions of two-dimensional orders. For width two orders a mixing rate of O(n3 logn) is proved. We conjecture that this is the mixing rate in the general case and support the conjecture by empirical data. On the course we obtain several nice structural results concerning Bruhat order and weak Bruhat order of permutations.