Euclidean Random Matching in 2D for Non-constant Densities
We consider the two-dimensional random matching problem in R^2 . In a challenging paper, Caracciolo et al. Phys Rev E 90(1):012118 (2014), on the basis of a subtle linearization of the Monge-Ampère 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 /2π N 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 /4π N .