Scatter n random points uniformly in a d-dimensional unit cube. A point is Pareto-optimal if no other point is better in every coordinate simultaneously. In low dimensions, most points are dominated — a few Pareto-optimal points sit on the frontier. In high dimensions, almost every point is Pareto-optimal — with enough criteria, everyone is best at something.

The transition between these regimes is sharp. There is a critical growth rate of d (relative to n) at which a phase transition occurs. Below the threshold, dominated points proliferate. Above it, they vanish. At the threshold, the count of dominated points converges to a Poisson distribution — the hallmark of a discrete phase transition where rare events of bounded multiplicity appear and disappear randomly.

The critical dimension scales as (2 ln n). This is the point where the combinatorial explosion of criteria overwhelms the statistical concentration of performance. Below 2 ln n dimensions, there is enough overlap in the criteria that some points dominate others. Above it, the criteria are sufficiently independent that dominance becomes impossible — every point has at least one coordinate where it leads.

What’s surprising is the universality for higher-order dominance. A point that dominates exactly r ≥ 2 others has a different critical dimension than the r = 0 or r = 1 case, but this critical dimension is the same for all r ≥ 2. The transition from “dominating two others” to “dominating none” happens at the same threshold regardless of the number dominated. The landscape of dominance simplifies abruptly: below the threshold, complex hierarchies exist; above it, the hierarchy collapses to a flat field.

(arXiv:2603.18698)