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 Hd(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 V D = [0, 1]d. For d ≥ 2, it is known that cd(n) = Hd(n)/n1/d converges in probability to a constant Cd < e, with Iim d→∞ Cd = e. For d = 2, the problem has been extensively studied, and it is known that C2 = 2; Cd is not currently known for any d ≥ 3. Straightforward Monte Carlo simulations to obtain Cd 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 Cd. We prove that Hd(n) can be estimated by considering only those chains close to the diagonal of the cube. A new conjecture regarding the asymptotic behavior of cd(n) leads to even more efficient experiments. We present experimental support for our conjecture, and the new estimates of Cd obtained from our experiments, for d ∈ {3,4,S,6}.