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Variational Problems with Percolation: Dilute Spin Systems at Zero Temperature

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Abstract

We study the asymptotic behavior of dilute spin lattice energies by exhibiting a continuous interfacial limit energy computed using the notion of Γ-convergence and techniques mixing Geometric Measure Theory and Percolation while scaling to zero the lattice spacing. The limit is not trivial above a percolation threshold. Since the lattice energies are not equi-coercive, a suitable notion of limit magnetization must be defined, which can be characterized by two phases separated by an interface. The macroscopic surface tension at this interface is characterized through a first-passage percolation formula, which highlights interesting connections between variational problems and percolation issues. A companion result on the asymptotic description on energies defined on paths in a dilute environment is also given.

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Acknowledgements

We acknowledge the valuable suggestions of the anonymous referee who pointed out Refs. [20] and [31], thanks to which we could improve our previous two-dimensional result.

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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. Department of Mathematics, Narvik University College, HiN, Postbox 385, 8505, Narvik, Norway

    Andrey Piatnitski

  3. P.N. Lebedev Physical Institute, RAS, 53 Leninski prospect, Moscow, 119991, Russia

    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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Braides, A., Piatnitski, A. Variational Problems with Percolation: Dilute Spin Systems at Zero Temperature. J Stat Phys 149, 846–864 (2012). https://doi.org/10.1007/s10955-012-0628-1

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  • DOI: https://doi.org/10.1007/s10955-012-0628-1

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