Inserted: 20 oct 2012
Last Updated: 6 dec 2012
Journal: J. Stat. Phys.
Links: paper page at J. Stat Phys
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