  We propose a new approach for the study of the quadratic stochastic Euclidean
bipartite matching problem between two sets of $N$ points each, $N\gg 1$. The
points are supposed independently randomly generated on a domain
$\Omega\subset\mathbb R^d$ with a given distribution $\rho(\mathbf x)$ on
$\Omega$. In particular, we derive a general expression for the correlation
function and for the average optimal cost of the optimal matching. A previous
ansatz for the matching problem on the flat hypertorus is obtained as
particular case.
