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Homogenization of Ferromagnetic Energies on Poisson Random Sets in the Plane

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Abstract

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.

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Acknowledgements

The authors acknowledge the MIUR Excellence Department Project awarded to the Department of Mathematics, University of Rome Tor Vergata, CUP E83C18000100006.

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Authors and Affiliations

  1. Dipartimento di Matematica, Università di Roma ‘Tor Vergata’, Via della Ricerca Scientifica, 00133, Rome, Italy

    Andrea Braides

  2. The Arctic University of Norway, UiT, Campus Narvik, P.O. Box 385, 8505, Narvik, Norway

    Andrey Piatnitski

  3. Institute for Information Transmission Problems of RAS, Moscow, Russia, 127051

    Andrey Piatnitski

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  1. Andrea Braides
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  2. Andrey Piatnitski
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Correspondence to Andrea Braides.

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Communicated by G. Dal Maso.

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Braides, A., Piatnitski, A. Homogenization of Ferromagnetic Energies on Poisson Random Sets in the Plane. Arch Rational Mech Anal 243, 433–458 (2022). https://doi.org/10.1007/s00205-021-01732-6

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