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 that is, 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.
MH is funded by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) under Germany’s Excellence Strategy EXC 2044-390685587, Mathematics Münster: Dynamics–Geometry–Structure and by the DFG through the SPP 2265 Random Geometric Systems. MG is partially funded by the ANR project SHAPO.
We warmly thank F. Otto for numerous discussions at various stages of the project. We also thank N. Berestycki and J. Aru for suggesting the connection between the standard GFF and the curl-free variant we consider (see Remark 2.2).
"A fluctuation result for the displacement in the optimal matching problem." Ann. Probab. 50 (4) 1446 - 1477, July 2022. https://doi.org/10.1214/21-AOP1562