*Published Paper*

**Inserted:** 20 oct 2012

**Last Updated:** 6 dec 2012

**Journal:** J. Stat. Phys.

**Volume:** 149

**Pages:** 846-864

**Year:** 2012

**Doi:** 10.1007/s10955-012-0628-1

**Links:**
paper page at J. Stat Phys

**Abstract:**

We study the asymptotic behaviour of dilute spin lattice energies by exhibiting a continuous interfacial limit energy computed using the notion of $\Gamma$-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.

**Keywords:**
Gamma-convergence, Dilute spins, lattice energies, first-passage percolation

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