  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.
