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

Understand

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.