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arXiv · 1504.05902

Onset of the Asymptotic Regime for Finite Orders

Joe Henson·David P. Rideout·Rafael D. Sorkin·Sumati Surya
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Abstract

We 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.

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Joe Henson, David P. Rideout, Rafael D. Sorkin, Sumati Surya. 2015-04-22. Onset of the Asymptotic Regime for Finite Orders. https://arxiv.org/abs/1504.05902

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