Calculus of Variations and Geometric Measure Theory

A. Braides - A. Piatnitski

Variational problems with percolation: dilute spin systems at zero temperature (the 2-dimensional case)

created by braidesa on 10 Feb 2011
modified on 23 Oct 2013


Unpublished manuscript

Inserted: 10 feb 2011
Last Updated: 23 oct 2013

Year: 2011


This is the 2-dimensional version of a paper with the same title. The proof presented here is at places simpler from the general case, and may be of independent interest.

We study the asymptotic behaviour of dilute spin lattice energies in dimension 2 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