Asymptotic behavior of the Dirichlet energy on Poisson point clouds

TL;DR

This paper demonstrates that quadratic pair interactions on scaled planar Poisson clouds can be almost surely approximated by a scaled Dirichlet energy, using a discrete-to-continuum limit and a coarse-grained convergence approach.

Contribution

It introduces a novel method to approximate pair interactions on Poisson clouds by Dirichlet energy through a discrete-to-continuum analysis.

Findings

Quadratic pair interactions are approximated by Dirichlet energy with a deterministic constant.

A compact discrete-to-continuum limit is established for scaled Poisson clouds.

A new coarse-grained convergence notion avoids exceptional regions in the Poisson cloud.

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.