Abstract
We consider a point cloud \(X_n := \{ {\mathbf {x}}_1, \ldots , {\mathbf {x}}_n \}\) 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 \(\varepsilon \) 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 in Chow et al. (Arch Ration Mech Anal 203:969–1008, 2012), Maas (J Funct Anal 261:2250–2292, 2011) and Mielke (Nonlinearity 24:1329–1346, 2011) for general finite Markov chains. We show that as long as \(\varepsilon = \varepsilon _n\) 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. 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.
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Acknowledgements
The author would like to thank Dejan Slepčev for enlightening discussions and for introducing him to the line of research investigated in this work. The author would also like to thank Jan Maas for enlightening discussions on this and other related topics. This manuscript was completed while the author was visiting the Erwin Schrödinger Institute to participate in the workshop “Optimal Transport: from Geometry to Numerics”. The author wants to thank the Institute for hospitality. Finally, the author would like to thank the anonymous referees for their suggestions.
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Communicated by M. Struwe.
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