Random matching in 2d: some recent results

I will consider the 2-dimensional random matching problem in two dimensional sets. In a challenging paper, Caracciolo et. al. on the basis of a subtle linearization of the Monge Ampere equation, conjectured that the expected value of the square of the Wasserstein distance, with exponent 2, between two samples of N uniformly distributed points in the unit square is logN / 2πN plus corrections.This and other related conjectures has been proved by Ambrosio et al. in a series of challenging papers.
In the talk I will review the results cited above and some related conjectures and proofs also in the case in which the density is non constant and positive in a bounded set. Also, I will discuss the case of densities defined on all the plane.