A large-scale regularity theory for the Monge-Ampere equation with rough data and application to the optimal matching problem

The aim of this paper is to obtain quantitative bounds for solutions to the optimal matching problem in dimension two. These bounds show that up to a logarithmically divergent shift, the optimal transport maps are close to be the identity at every scale. These bounds allow us to pass to the limit as the system size goes to infinity and construct a locally optimal coupling between the Lebesgue measure and the Poisson point process which retains the stationarity properties of the Poisson point process only at the level of second-order differences. Our quantitative bounds are obtained through a Campanato iteration scheme based on a deterministic and a stochastic ingredient. The deterministic part, which can be seen as our main contribution, is a regularity result for Monge-Amp\`ere equations with rough right-hand side. Since we believe that it could be useful in other contexts, we prove it for general space dimensions. The stochastic part is a concentration result for the optimal matching problem which builds on previous work by Ambrosio, Stra and Trevisan.