Title: Gromov-Hausdorff limit of Wasserstein spaces on point clouds,
Abstract: We consider a point cloud $X_n := \{ x_1, \dots, x_n \}$ uniformly
distributed on the flat torus $\mathbb{T}^d : = \mathbb{R}^d / \mathbb{Z}^d $,
and construct a geometric graph on the cloud by connecting points that are
within distance $\epsilon$ of each other. We let $\mathcal{P}(X_n)$ be the
space of probability measures on $X_n$ and endow it with a discrete Wasserstein
distance $W_n$ as introduced independently by Maas and Zhou et al. for general
finite Markov chains. We show that as long as $\epsilon= \epsilon_n$ decays
towards zero slower than an explicit rate depending on the level of uniformity
of $X_n$, then the space $(\mathcal{P}(X_n), W_n)$ converges in the
Gromov-Hausdorff sense towards the space of probability measures on
$\mathbb{T}^d$ endowed with the Wasserstein distance.