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Random high-dimensional orders
Published online by Cambridge University Press: 01 July 2016
Abstract
The random k-dimensional partial order P k (n) on n points is defined by taking n points uniformly at random from [0,1]k . Previous work has concentrated on the case where k is constant: we consider the model where k increases with n.
We pay particular attention to the height H k (n) of P k (n). We show that k = (t/log t!) log n is a sharp threshold function for the existence of a t-chain in P k (n): if k – (t/log t!) log n tends to + ∞ then the probability that P k (n) contains a t-chain tends to 0; whereas if the quantity tends to − ∞ then the probability tends to 1. We describe the behaviour of H k (n) for the entire range of k(n).
We also consider the maximum degree of P k (n). We show that, for each fixed d ≧ 2, 
is a threshold function for the appearance of an element of degree d. Thus the maximum degree undergoes very rapid growth near this value of k.
We make some remarks on the existence of threshold functions in general, and give some bounds on the dimension of P k (n) for large k(n).
Keywords
MSC classification
Primary:
60C05: Combinatorial probability
Secondary:
06A06: Partial order, general
Information
- Type
- General Applied Probability
- Information
- Copyright
- Copyright © Applied Probability Trust 1995
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