Quadratic stochastic Euclidean bipartite matching problem

TL;DR

This paper introduces a new analytical approach to the quadratic stochastic Euclidean bipartite matching problem, deriving general expressions for correlation functions and average costs, applicable to large random point sets in various domains.

Contribution

The paper develops a novel method for analyzing the quadratic stochastic Euclidean bipartite matching problem, extending previous results and providing general formulas for correlation functions and costs.

Findings

Derived a general expression for the correlation function.

Calculated the average optimal matching cost.

Extended previous ansatz to more general domains.

Abstract

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.