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November, 1992 The Height of a Random Partial Order: Concentration of Measure
Bela Bollobas, Graham Brightwell
Ann. Appl. Probab. 2(4): 1009-1018 (November, 1992). DOI: 10.1214/aoap/1177005586


The problem of determining the length $L_n$ of the longest increasing subsequence in a random permutation of $\{1, \ldots, n\}$ is equivalent to that of finding the height of a random two-dimensional partial order (obtained by intersecting two random linear orders). The expectation of $L_n$ is known to be about $2\sqrt{n}$. Frieze investigated the concentration of $L_n$ about this mean, showing that, for $\varepsilon > 0$, there is some constant $\beta > 0$ such that $Pr(|L_n - \mathbf{E}L_n| \geq n^{1/3+\varepsilon}) \leq \exp(-n^\beta).$ In this paper we obtain similar concentration results for the heights of random $k$-dimensional orders, for all $k \geq 2$. In the case $k = 2$, our method replaces the $n^{1/3+\varepsilon}$ above with $n^{1/4+\varepsilon}$, which we believe to be essentially best possible.


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Bela Bollobas. Graham Brightwell. "The Height of a Random Partial Order: Concentration of Measure." Ann. Appl. Probab. 2 (4) 1009 - 1018, November, 1992.


Published: November, 1992
First available in Project Euclid: 19 April 2007

zbMATH: 0758.06001
MathSciNet: MR1189428
Digital Object Identifier: 10.1214/aoap/1177005586

Primary: 06A10
Secondary: 05A99 , 60C05

Keywords: Height , increasing subsequences , partial order , random orders , Ulam's problem

Rights: Copyright © 1992 Institute of Mathematical Statistics

Vol.2 • No. 4 • November, 1992
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