Gromov-Hausdorff convergence of discrete transportation metrics

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 W_2 on the continuous torus in the sense of Gromov-Hausdorff. This is the first convergence result for the recently developed discrete transportation metrics. The result shows the compatibility between these metrics and the well-established 2-Wasserstein metric.