  The aim of this paper is to justify in dimensions two and three the ansatz of
Caracciolo et al. stating that the displacement in the optimal matching problem
is essentially given by the solution to the linearized equation i.e. the
Poisson equation. Moreover, we prove that at all mesoscopic scales, this
displacement is close in suitable negative Sobolev spaces to a curl-free
Gaussian free field. For this we combine a quantitative estimate on the
difference between the displacement and the linearized object, which is based
on the large-scale regularity theory recently developed in collaboration with
F. Otto, together with a qualitative convergence result for the linearized
problem.
