# On the optimal map in the 2-dimensional random matching problem

## TL;DR

This paper demonstrates that on a 2D compact manifold, the optimal transport map in a semi-discrete random matching problem can be effectively approximated by a simple analytical form involving the solution to a Poisson equation, confirming a scaling hypothesis.

## Contribution

It establishes a rigorous approximation of the optimal transport map using the Poisson problem, extending previous hypotheses to include the map's behavior on manifolds.

## Key findings

- Optimal map approximated by identity plus gradient of Poisson solution
- Validation of Caracciolo et al.'s scaling hypothesis for the map
- New stability result for optimal transport maps on manifolds

## Abstract

We show that, on a $2$-dimensional compact manifold, the optimal transport map in the semi-discrete random matching problem is well-approximated in the $L^2$-norm by identity plus the gradient of the solution to the Poisson problem $-\Delta f^{n,t} = \mu^{n,t}-1$, where $\mu^{n,t}$ is an appropriate regularization of the empirical measure associated to the random points. This shows that the ansatz of Caracciolo et al. (Scaling hypothesis for the Euclidean bipartite matching problem) is strong enough to capture the behavior of the optimal map in addition to the value of the optimal matching cost. As part of our strategy, we prove a new stability result for the optimal transport map on a compact manifold.