## The Annals of Applied Probability

- Ann. Appl. Probab.
- Volume 2, Number 4 (1992), 1009-1018.

### The Height of a Random Partial Order: Concentration of Measure

Bela Bollobas and Graham Brightwell

#### Abstract

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.

#### Article information

**Source**

Ann. Appl. Probab., Volume 2, Number 4 (1992), 1009-1018.

**Dates**

First available in Project Euclid: 19 April 2007

**Permanent link to this document**

https://projecteuclid.org/euclid.aoap/1177005586

**Digital Object Identifier**

doi:10.1214/aoap/1177005586

**Mathematical Reviews number (MathSciNet)**

MR1189428

**Zentralblatt MATH identifier**

0758.06001

**JSTOR**

links.jstor.org

**Subjects**

Primary: 06A10

Secondary: 60C05: Combinatorial probability 05A99: None of the above, but in this section

**Keywords**

Partial order height random orders Ulam's problem increasing subsequences

#### Citation

Bollobas, Bela; Brightwell, Graham. The Height of a Random Partial Order: Concentration of Measure. Ann. Appl. Probab. 2 (1992), no. 4, 1009--1018. doi:10.1214/aoap/1177005586. https://projecteuclid.org/euclid.aoap/1177005586