Homogenization of Ferromagnetic Energies on Poisson Random Sets in the Plane

Abstract

We prove that by scaling nearest-neighbour ferromagnetic energies de ned 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 de ne a compact convergence, and a characterisation of metric properties of clusters of Voronoi cells using limit theorems for subadditive processes.