Calculus of Variations and Geometric Measure Theory

A. Braides - M. Caroccia

Asymptotic behavior of the Dirichlet energy on Poisson point clouds

created by braidesa on 30 Mar 2022
modified on 13 Jul 2023


Published Paper

Inserted: 30 mar 2022
Last Updated: 13 jul 2023

Journal: J. Nonlinear Sci.
Volume: 33
Number: 80
Year: 2023
Doi: 10.1007/s00332-023-09937-7
Links: Journal read-only version


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.

Keywords: Homogenization, percolation, discrete-to-continuum, random sets