The Most Dispersed Subset of Random Points in $\mathbb{R}^d$

Consider a population of $N$ individuals, each having $d\geq 1$ different traits, and an additive measure, called dispersion, which rewards large pairwise separations between traits. The goal is to select $M\leq N$ individuals such that their traits are as dispersed as possible. We compute analytically the full statistics (including large deviation tails) of the maximally achievable dispersion among sub-populations of size $M$ when the traits are independent and identically distributed. Two complementary approaches are developed, one based on a mean-field theory for order statistics, and the other on the replica method from the field of disordered systems. In all dimensions $d$, and for rotationally symmetric distributions, the optimal subset for large populations consists of all points lying outside a $d$-dimensional ball whose radius is determined self-consistently. For a single trait ($d=1$), the statistics of the maximal dispersion can be tackled for finite $N,M$ as well. The formulae we obtained are corroborated by numerical simulations on small instances and by heuristic algorithms that find near-optimal solutions.