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The aim of this paper is to obtain quantitative bounds for solutions to theoptimal matching problem in dimension two. These bounds show that up to alogarithmically divergent shift, the optimal transport maps are close to be theidentity at every scale. These bounds allow us to pass to the limit as thesystem size goes to infinity and construct a locally optimal coupling betweenthe Lebesgue measure and the Poisson point process which retains thestationarity properties of the Poisson point process only at the level ofsecond-order differences. Our quantitative bounds are obtained through aCampanato iteration scheme based on a deterministic and a stochasticingredient. 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 forgeneral space dimensions. The stochastic part is a concentration result for theoptimal matching problem which builds on previous work by Ambrosio, Stra andTrevisan.

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