  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.
