Asymptotic behavior of the dirichlet energy on poisson point clouds
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Date
2022-03-23
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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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SISSA 06/2022/MATE