Pareto points in growing dimensions

TL;DR

This paper studies the behavior of Pareto points among random points in high-dimensional cubes, identifying phase transitions in their number and distribution as the dimension grows, with implications for understanding dominance relations.

Contribution

It determines the critical growth rate of dimension where phase transitions occur for Pareto points and their dominance properties, revealing new asymptotic behaviors.

Findings

Number of non-Pareto points diverges below critical dimension

Poisson distribution of non-Pareto points at criticality

Distinct phase transition thresholds for points dominating exactly r others

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 $r$.