A PDE approach to a 2-dimensional matching problem

TL;DR

This paper develops a PDE-based method to analyze the asymptotic behavior of quadratic transportation costs in 2D random matching problems, providing leading-term estimates for empirical measures of uniform samples.

Contribution

It introduces a rigorous PDE formulation to derive asymptotic results for 2D matching problems, extending previous heuristic approaches.

Findings

Derived the leading term in the asymptotic expansion of expected transportation cost.

Established a PDE framework to analyze 2D matching problems.

Validated the approach with rigorous mathematical proofs.

Abstract

We prove asymptotic results for 2-dimensional random matching problems. In particular, we obtain the leading term in the asymptotic expansion of the expected quadratic transportation cost for empirical measures of two samples of independent uniform random variables in the square. Our technique is based on a rigorous formulation of the challenging PDE ansatz by S.\ Caracciolo et al.\ (Phys. Rev. E, {\bf 90} 012118, 2014) that "linearise" the Monge-Amp\`ere equation.