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

TL;DR

This paper derives the two-point correlation function for the optimal matching in the Euclidean bipartite matching problem on a hypertorus, linking it to the Laplace operator's Green's function, and extends the analysis to different matching scenarios.

Contribution

It provides a detailed derivation of the correlation function for the bipartite matching problem using a novel ansatz, connecting it to the Laplace operator's Green's function.

Findings

Correlation function explicitly derived for the hypertorus.

Connection established between correlation function and Laplace operator's Green's function.

Analysis covers both grid-Poisson and Poisson-Poisson matching problems.

Abstract

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.