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arXiv · 2203.16877

Asymptotic behavior of the Dirichlet energy on Poisson point clouds

Andrea Braides·Marco Caroccia
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Abstract

We prove that quadratic pair interactions for functions defined on planar Poisson clouds and taking into account pairs of sites of distance up to a certain (large-enough) threshold can be almost surely approximated by the multiple of the Dirichlet energy by a deterministic constant. This is achieved by scaling the Poisson cloud and the corresponding energies and computing a compact discrete-to-continuum limit. In order to avoid the effect of exceptional regions of the Poisson cloud, with an accumulation of sites or with "disconnected sites", a suitable "coarse-grained" notion of convergence of functions defined on scaled Poisson clouds must be given.

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Andrea Braides, Marco Caroccia. 2022-03-31. Asymptotic behavior of the Dirichlet energy on Poisson point clouds. https://doi.org/10.1007/s00332-023-09937-7

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