Finer estimates on the 2-dimensional matching problem

TL;DR

This paper refines the understanding of the expected cost in the 2D random matching problem on manifolds, providing simplified proofs, precise asymptotic coefficients, and improved error estimates, with new technical tools for heat flow analysis.

Contribution

It offers a simplified proof, determines the leading coefficient of the asymptotic expansion, and sharpens error bounds for the 2D matching problem on manifolds, introducing a refined heat flow estimate.

Findings

Exact asymptotic coefficient for expected matching cost

Simplified proof approach for the asymptotic behavior

Improved error term estimates for semi-discrete matching

Abstract

We study the asymptotic behaviour of the expected cost of the random matching problem on a $2$-dimensional compact manifold, improving in several aspects the results of L. Ambrosio, F. Stra and D. Trevisan (A PDE approach to a 2-dimensional matching problem). In particular, we simplify the original proof (by treating at the same time upper and lower bounds) and we obtain the coefficient of the leading term of the asymptotic expansion of the expected cost for the random bipartite matching on a general 2-dimensional closed manifold. We also sharpen the estimate of the error term given by M. Ledoux (On optimal matching of Gaussian samples II) for the semi-discrete matching. As a technical tool, we develop a refined contractivity estimate for the heat flow on random data that might be of independent interest.