We study the asymptotic behavior of dilute spin lattice energies by exhibiting a continuous interfacial limit energy computed using the notion of I"-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.
Braides, A., Piatnitski, A. (2012). Variational Problems with Percolation: Dilute Spin Systems at Zero Temperature. JOURNAL OF STATISTICAL PHYSICS, 149(5), 846-864 [10.1007/s10955-012-0628-1].
Variational Problems with Percolation: Dilute Spin Systems at Zero Temperature
BRAIDES, ANDREA;
2012-01-01
Abstract
We study the asymptotic behavior of dilute spin lattice energies by exhibiting a continuous interfacial limit energy computed using the notion of I"-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.File | Dimensione | Formato | |
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