On the one dimensional Euclidean matching problem: exact solutions, correlation functions and universality

TL;DR

This paper explores the Euclidean bipartite matching problem on a line and circle, establishing equivalences with Brownian bridge processes to compute correlation functions and optimal costs in the thermodynamic limit.

Contribution

It introduces an equivalence between the matching problem and Brownian bridges, enabling exact solutions for correlation functions and costs, including minimax problems.

Findings

Correlation functions are explicitly computed.

Optimal costs are derived in the thermodynamic limit.

Solutions for minimax problems are provided.

Abstract

We discuss the equivalence relation between the Euclidean bipartite matching problem on the line and on the circumference and the Brownian bridge process on the same domains. The equivalence allows us to compute the correlation function and the optimal cost of the original combinatoric problem in the thermodynamic limit; moreover, we solve also the minimax problem on the line and on the circumference. The properties of the average cost and correlation functions are discussed.