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 wellapproximated 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.