Leírás
Abstract:
Ajtai, Komlós, and Tusnády studied the optimal matching between two large clouds of independently and uniformly distributed points on the torus. They focused on the critical dimension two, where they showed that the typical transportation distance is logarithmically larger than the typical distance between the points. When the cost functional is the quadratic distance, the matching is cyclically monotone; when the size of the torus tends to infinity, the points are distributed according to the Poisson point process. We show that there is no cyclically monotone and invariant (i. e. stationary) matching between two independent copies of the Poisson point process.
The argument relies on a recently developed purely variational regularity theory for optimal transportation. More specifically, it relies on the observation that in certain regimes, the displacement is well-approximated by the gradient of a harmonic function. This approach to epsilon-regularity is reminiscent of de Giorgi’s approach to regularity theory for minimal surfaces, which is based on the approximation by the graph of a harmonic function. It is orthogonal to Caffarelli’s approach to regularity for optimal transportation, which capitalizes on the maximum principle for the Monge-Ampère equation.
This is joint work with M. Huesmann and F. Mattesini, relying on earlier work with M. Goldman.