Pareto points in growing dimensions

Abstract

We consider $n$ independent random points uniformly distributed in the $d_n$-dimensional unit cube and study Pareto points, that is, points that do not coordinatewise dominate any other point. We identify the critical growth rate of $d_n$ at which a phase transition occurs: below this threshold, the number of non-Pareto points diverges in probability, whereas above it there are asymptotically no such points. At criticality, the number of non-Pareto points converges in distribution to a Poisson random variable. We further describe their asymptotic spatial distribution in terms of convergence of random point measures. We also investigate points that dominate exactly $r$ other points and establish analogous phase transitions. For $r=1$, the critical dimension is the same as for non-Pareto points, whereas for every fixed $r\geq 2$ it is different, but, surprisingly, common to all such $r$.