Convergence of asymptotic costs for random Euclidean matching problems

TL;DR

This paper proves that the average minimum cost of bipartite Euclidean matching between random points converges to a positive constant as the number of points grows, for dimensions d ≥ 3 and cost powers p ≥ 1.

Contribution

It establishes the convergence of the asymptotic matching cost in higher dimensions and extends PDE-based methods to analyze these problems.

Findings

Convergence of normalized matching costs to a positive constant in dimensions d ≥ 3.

Extension of PDE methods to higher-dimensional matching problems.

Analysis of optimal transport between random points and the uniform measure.

Abstract

We investigate the average minimum cost of a bipartite matching between two samples of n independent random points uniformly distributed on a unit cube in d $\ge$ 3 dimensions, where the matching cost between two points is given by any power p $\ge$ 1 of their Euclidean distance. As n grows, we prove convergence, after a suitable renormalization, towards a finite and positive constant. We also consider the analogous problem of optimal transport between n points and the uniform measure. The proofs combine sub-additivity inequalities with a PDE ansatz similar to the one proposed in the context of the matching problem in two dimensions and later extended to obtain upper bounds in higher dimensions.