CONVERGENCE OF ASYMPTOTIC COSTS FOR RANDOM EUCLIDEAN MATCHING PROBLEMS

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 ≥ 3 dimensions, where the matching cost between two points is given by any power p ≥ 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 subadditivity 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.