Homogenization of Ferromagnetic Energies on Poisson Random Sets in the Plane

We prove that by scaling nearest-neighbour ferromagnetic energies defined on Poisson random sets in the plane one obtains 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.

Automated summary

By scaling Poisson random sets in the plane, we can obtain an isotropic perimeter energy with an asymptotic formula, which can be achieved by neglecting cells with very long or very short edges in the Voronoi model. Using tools from Geometry Measure Theory and limit theorems, we can define compact convergence and characterize the metric properties of clusters of Voronoi cells.