ABSTRACTWe describe a Markov Chain Monte Carlo algorithm which can be used to generate naturally labeled n-element posets at random with a probability distribution of one’s choice. Implementing this algorithm for the uniform distribution, we explore the approach to the asymptotic regime in which almost every poset takes on the three-layer structure described by Kleitman and Rothschild (KR). By tracking the n-dependence of several order-invariants, among them the height of the poset, we observe an oscillatory behavior which is very unlike a monotonic approach to the KR regime. Only around n = 40 or so does this “finite size dance” appear to give way to a gradual crossover to asymptopia which lasts until n = 85, the largest n we have simulated.
['Joe Henson', 'David Rideout', 'Rafael D. Sorkin', 'Sumati Surya']
Onset of the Asymptotic Regime for (Uniformly Random) Finite Orders