Calculus of Variations and Geometric Measure Theory

L. Ambrosio - M. Goldman - D. Trevisan

On the quadratic random matching problem in two-dimensional domains

created by goldman on 27 Oct 2021
modified by trevisan on 05 Jul 2024


Accepted Paper

Inserted: 27 oct 2021
Last Updated: 5 jul 2024

Journal: Electronic J. Probability
Year: 2021

ArXiv: 2110.14372 PDF


We investigate the average minimum cost of a bipartite matching, with respect to the squared Euclidean distance, between two samples of n i.i.d. random points on a bounded Lipschitz domain in the Euclidean plane, whose common law is absolutely continuous with strictly positive H{\"o}lder continuous density. We confirm in particular the validity of a conjecture by D. Benedetto and E. Caglioti stating that the asymptotic cost as n grows is given by the logarithm of n multiplied by an explicit constant times the volume of the domain. Our proof relies on a reduction to the optimal transport problem between the associated empirical measures and a Whitney-type decomposition of the domain, together with suitable upper and lower bounds for local and global contributions, both ultimately based on PDE tools. We further show how to extend our results to more general settings, including Riemannian manifolds, and also give an application to the asymptotic cost of the random quadratic bipartite travelling salesperson problem.