# Annealed quantitative estimates for the quadratic 2D-discrete random matching problem

## TL;DR

This paper analyzes the quadratic 2D-discrete random matching problem on Riemannian manifolds, providing quantitative approximations of optimal transport plans for correlated and Markov chain samples.

## Contribution

It introduces a novel approximation method for optimal transport plans using PDE linearization, applicable to correlated samples and Markov chains.

## Key findings

- Optimal transport plans are well-approximated by PDE-based maps.
- The method applies to correlated random points with exponential decay of mixing coefficients.
- Results extend to discrete-time Markov chains with invariant measures.

## Abstract

We study a random matching problem on closed compact $2$-dimensional Riemannian manifolds (with respect to the squared Riemannian distance), with samples of random points whose common law is absolutely continuous with respect to the volume measure with strictly positive and bounded density. We show that given two sequences of numbers $n$ and $m=m(n)$ of points, asymptotically equivalent as $n$ goes to infinity, the optimal transport plan between the two empirical measures $\mu^n$ and $\nu^{m}$ is quantitatively well-approximated by $\big(\mathrm{Id},\exp(\nabla h^{n})\big)_\#\mu^n$ where $h^{n}$ solves a linear elliptic PDE obtained by a regularized first-order linearization of the Monge-Amp\`ere equation. This is obtained in the case of samples of correlated random points for which a stretched exponential decay of the $\alpha$-mixing coefficient holds and for a class of discrete-time Markov chains having a unique absolutely continuous invariant measure with respect to the volume measure.