Calculus of Variations and Geometric Measure Theory

A. Braides - A. Piatnitski

Homogenization of surface and length energies for spin systems

created by braidesa on 30 Jun 2010
modified on 09 Feb 2013


Published Paper

Inserted: 30 jun 2010
Last Updated: 9 feb 2013

Journal: J. Funct. Anal.
Volume: 264
Pages: 1296-1328
Year: 2013


We study the homogenization of lattice energies related to Ising systems of the form

$E_\varepsilon(u)=-\sum_{ij} c^\varepsilon_{ij} u_i u_j,$

with $u_i$ a spin variable indexed on the portion of a cubic lattice $\Omega\cap\varepsilon \mathbb Z^d$, by computing their $\Gamma$-limit in the framework of surface energies in a BV setting. We introduce a notion of homogenizability of the system $\{c^\varepsilon_{ij}\}$ that allows to treat periodic, almost-periodic and random statistically homogeneous models, when the coefficients are positive (ferromagnetic energies), in which case the limit energy is finite on $BV(\Omega;\{\pm1\})$ and takes the form

$ F(u)=\int_{\Omega\cap\partial^*\{u=1\}}\varphi(\nu)d{\mathcal H}^{d-1} $

($\nu$ is the normal to $\partial^*\{u=1\}$), where $\varphi$ is characterized by an asymptotic formula. In the random case $\varphi$ can be expressed in terms of first-passage percolation characteristics. The result is extended to coefficients with varying sign, under the assumption that the areas where the energies are antiferromagnetic are well separated. Finally, we prove a dual result for discrete curves.

Keywords: Gamma-convergence, surface energies, spin systems, Discrete-to-continuous homogenization