Scaling hypothesis for the Euclidean bipartite matching problem

We propose a simple yet very predictive form, based on a Poisson's equation, for the functional dependence of the cost from the density of points in the Euclidean bipartite matching problem. This leads, for quadratic costs, to the analytic prediction of the large N limit of the average cost in dimension d=1,2 and of the subleading correction in higher dimension. A nontrivial scaling exponent, γ_d=d-2/d, which differs from the monopartite's one, is found for the subleading correction. We argue that the same scaling holds true for a generic cost exponent in dimension d>2.