Scaling hypothesis for the Euclidean bipartite matching problem II. Correlation functions

We analyze the random Euclidean bipartite matching problem on the hypertorus in $d$ dimensions with quadratic cost and we derive the two–point correlation function for the optimal matching, using a proper ansatz introduced by Caracciolo et al. to evaluate the average optimal matching cost. We consider both the grid–Poisson matching problem and the Poisson–Poisson matching problem. We also show that the correlation function is strictly related to the Green’s function of the Laplace operator on the hypertorus.