Local and non local continuum limits of Ising type energies for spin systems

created by alicandr on 31 Jul 2014
modified on 23 Mar 2016

[BibTeX]

Published Paper

Inserted: 31 jul 2014
Last Updated: 23 mar 2016

Journal: SIAM J. Math. Anal.
Volume: 48
Number: 2
Pages: 895-931
Year: 2016

Abstract:

We study, through a $\Gamma$-convergence procedure, the discrete to continuum limit of Ising type energies of the form $F_\varepsilon (u)=-\sum_{i,j}c_{i,j}^\varepsilon u_i u_j,$ where $u$ is a spin variable defined on a portion of a cubic lattice $\varepsilon {\Bbb Z}^d\cap\Omega$, $\Omega$ being a regular bounded open set, and valued in $\{-1,1\}$. If the constants $c_{i,j}^\varepsilon$ are non negative and satisfy suitable coercivity and decay assumptions, we show that all possible $\Gamma$-limits of surface scalings of the functionals $F_\varepsilon$ are finite on $BV(\Omega;\{-1,1\}$ and of the form $\int_{S_u}\varphi(x,\nu_u)\, d{\cal H}^{d-1}.$ If such decay assumptions are violated, we show that we may approximate non local functionals of the form $\int_{S_u}\varphi(\nu_u)\, d{\cal H}^{d-1}+\int_\Omega\int_\Omega K(x,y)g(u(x),u(y))\, dxdy.$ We focus on the approximation of two relevant examples: fractional perimeters and Ohta-Kawasaki type energies. Eventually, we provide a general criterion for a ferromagnetic behavior of the energies $F_\varepsilon$ even when the constants $c_{i,j}^\varepsilon$ change sign. If such criterion is satisfied, the ground states of $F_\varepsilon$ are still the uniform states $1$ and $-1$ and the continuum limit of the scaled energies is an integral surface energy of the form above.