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Random high-dimensional orders

Part of: Ordered sets Combinatorial probability

Published online by Cambridge University Press:  01 July 2016

Béla Bollobás  and
Graham Brightwell
Béla Bollobás*
Affiliation:
University of Cambridge
Graham Brightwell*
Affiliation:
London School of Economics and Political Science
*
* Postal address: Department of Pure Mathematics and Mathematical Statistics, University of Cambridge, 16 Mill Lane, Cambridge, CB2 1SB, UK.
** Postal address: Department of Mathematics, London School of Economics and Political Science, Houghton Street, London, WC2A 2AE, UK.
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Abstract

The random k-dimensional partial order Pk(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 Hk(n) of Pk(n). We show that k = (t/log t!) log n is a sharp threshold function for the existence of a t-chain in Pk(n): if k – (t/log t!) log n tends to + ∞ then the probability that Pk(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 Hk(n) for the entire range of k(n).

We also consider the maximum degree of Pk(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 Pk(n) for large k(n).

Keywords

RANDOM PARTIAL ORDERORDER DIMENSION

MSC classification

Primary: 60C05: Combinatorial probability
Secondary: 06A06: Partial order, general
Type
General Applied Probability
Information
Advances in Applied Probability , Volume 27 , Issue 1 , March 1995 , pp. 161 - 184
Copyright
Copyright © Applied Probability Trust 1995 

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