The Height of a Random Partial Order: Concentration of Measure

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.