On the Height of a Random Set of Points in a $d$-Dimensional Unit Cube

Abstract

We investigate, through numerical experiments, the asymptotic behavior of the length $H_d(n)$ of a maximal chain (longest totally ordered subset) of a set of $n$ points drawn from a uniform distribution on the $d$-dimensional unit cube {\bf V}\math{_D=[0,1]^d}. For \math{d\ge2}, it is known that \math{c_d(n)=H_d(n)/n^{1/d}} converges in probability to a constant \math{c_d<e}, with \math{\lim_{d\rightarrow\infty} c_d=e}. For \math{d=2}, the problem has been extensively studied, and it is known that \math{c_2=2}; \math{c_d} is not currently known for any \math{d\ge3}. Straightforward Monte Carlo simulations to obtain \math{c_d} have already been proposed, and shown to be beyond the scope of current computational resources. In this paper, we present a computational approach which yields feasible experiments that lead to estimates for \math{c_d}. We prove that \math{H_d(n)} can be estimated by considering only those chains close to the diagonal of the cube. A new conjecture regarding the asymptotic behavior of \math{c_d(n)} leads to even more efficient experiments. We present experimental support for our conjecture, and the new estimates of \math{c_d} obtained from our experiments, for \math{d\in\{3,4,5,6\}}.