A fluctuation result for the displacement in the optimal matching problem

TL;DR

This paper confirms that in 2D and 3D, the displacement in the optimal matching problem behaves like the solution to a linear Poisson equation and resembles a curl-free Gaussian free field at mesoscopic scales.

Contribution

It provides a rigorous justification of the linearized ansatz for displacement and characterizes its Gaussian free field nature at mesoscopic scales in low dimensions.

Findings

Displacement approximates the solution to the Poisson equation.

Displacement resembles a curl-free Gaussian free field.

Quantitative estimates link displacement to linearized models.

Abstract

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.