Gromov-Hausdorff convergence of discrete transportation metrics
Résumé
This paper continues the investigation of 'Wasserstein-like' transportation distances for probability measures on discrete sets. We prove that the discrete transportation metrics on the d-dimensional discrete torus with mesh size 1/N converge, when $N\to\infty$, to the standard 2-Wasserstein distance on the continuous torus in the sense of Gromov-Hausdorff. This is the first result of a passage to the limit from a discrete transportation problem to a continuous one, and proves compatibility of the recently developed discrete metrics and the well-established 2-Wasserstein metric.