Convergence of asymptotic costs for random Euclidean matching problems

Michael Goldman (1) , Dario Trevisan (2)

1 LJLL - Laboratoire Jacques-Louis Lions
2 University of Pisa - Università di Pisa

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 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.

Domaines: Probabilités [math.PR]; Equations aux dérivées partielles [math.AP]; Physique mathématique [math-ph]

Michael Goldman, Dario Trevisan. Convergence of asymptotic costs for random Euclidean matching problems. Probability and Mathematical Physics, In press. ⟨hal-02932968v2⟩