*Preprint*

**Inserted:** 24 jan 2020

**Last Updated:** 24 jan 2020

**Year:** 2020

**Notes:**

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

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