Optimal matching and balancing transport
The optimal matching problem is one of the classical problems in probability. By now, there is a good understanding of the macroscopic behaviour with some very detailed results, several challenging predictions, and open problems. The goal of this project is to develop a refined analysis of solutions to the optimal matching problem from a macroscopic scale down to a microscopic scale. Based on a recent quantitative linearization result of the Monge-Ampère equation developed in collaboration with Michael Goldman, we will investigate two main directions. On the one hand, we aim at rigorously connecting the solutions to the optimal matching problem to a Gaussian field which scales as the Gaussian free field. On the other hand, we seek to close the gap and show that rescaled solutions to the optimal matching problem converge in the thermo-dynamic limit to invariant allocations to point processes, or more generally to balancing transports between random measures. In the long run, combining these limit results with the Gaussian like behaviour of solutions to the matching problem, we seek to analyse the random tessellations induced by the limiting invariant allocations.
Michael Goldman, Martin Huesmann: A fluctuation result for the displacement in the optimal matching problem (05/2021)