Gromov-Hausdorff limit of Wasserstein spaces on point clouds

We consider a point cloud Xn := x1, dots, xn uniformly distributed on the flat torus mathbbTd : = ℝd / mathbbZd , and construct a geometric graph on the cloud by connecting points that are within distance varepsilon of each other. We let P(Xn) be the space of probability measures on Xn and endow it with a discrete Wasserstein distance Wn as introduced independently by Chow et al, Maas, and Mielke for general finite Markov chains. We show that as long as varepsilon= varepsilonn decays towards zero slower than an explicit rate depending on the level of uniformity of Xn, then the space (P(Xn), Wn) converges in the Gromov-Hausdorff sense towards the space of probability measures on mathbbTd endowed with the Wasserstein distance. The analysis presented in this paper is a first step in the study of stability of evolution equations defined over random point clouds as the number of points grows to infinity.