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

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 [8] 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.