Euclidean random matching in 2D for non-constant densities

We consider the 2-dimensional random matching problem in $\mathbb{R}^2.$ In a challenging paper, Caracciolo et. al. arXiv:1402.6993 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 $\log N/2πN$ plus corrections, while the expected value of the square of the Wasserstein distance between one sample of $N$ uniformly distributed points and the uniform measure on the square is $\log N/4πN$. These conjectures has been proved by Ambrosio et al. arXiv:1611.04960. Here we consider the case in which the points are sampled from a non uniform density. For first we give formal arguments leading to the conjecture that if the density is regular and positive in a regular, bounded and connected domain $Λ$ in the plane, then the leading term of the expected values of the Wasserstein distances are exactly the same as in the case of uniform density, but for the multiplicative factor equal to the measure of $Λ$. We do not prove these results but, in the case in which the domain is a square, we prove estimates from above that coincides with the conjectured result.

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