A fluctuation result for the optimal matching problem

The optimal matching problem is one of the classical random optimization problems. While the asymptotic behavior of the expected cost is well understood only little is known for the asymptotic behavior of the optimal couplings - the solutions to the optimal matching problem. In this talk we show that at all mesoscopic scales the displacement under the optimal coupling converges in suitable Sobolev spaces to a Gaussian field which can be identified as the curl-free part of a vector Gaussian free field. Based on joint work with Michael Goldman.