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

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 L2-norm by identity plus the gradient of the solution to the Poisson problem −∆fn,t = µn,t − 1, where µn,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.