Calculus of Variations and Geometric Measure Theory

A. Braides - A. Piatnitski

Homogenization of ferromagnetic energies on Poisson random sets in the plane

created by braidesa on 24 Jan 2020
modified on 21 Jan 2022


Published Paper

Inserted: 24 jan 2020
Last Updated: 21 jan 2022

Journal: Arch. Rational Mech. Anal.
Volume: 243
Pages: 433-458
Year: 2022
Doi: 10.1007/s00205-021-01732-6

We prove that by scaling nearest-neighbour ferromagnetic energies defined on Poisson random sets in the plane we obtain an isotropic perimeter energy with a surface tension characterised by an asymptotic formula. The result relies on proving that cells with 'very long' or 'very short' edges of the corresponding Voronoi tessellation can be neglected. In this way we may apply Geometry Measure Theory tools to define a compact convergence, and a characterisation of metric properties of clusters of Voronoi cells using limit theorems for subadditive processes.

Keywords: Homogenization, discrete systems, first-passage percolation, Interfacial energies, Poisson random sets, ferromagnetic energies