Calculus of Variations and Geometric Measure Theory

A. Braides - A. Piatnitski

Variational problems with percolation: dilute spin systems at zero temperature

created by braidesa on 20 Oct 2012
modified on 06 Dec 2012


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


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