Local and nonlocal continuum limits of Ising-type energies for spin systems (Q2793834)

Based on the authors' abstract: In this article, the discrete to continuum limit of Ising-type energies of the form NEWLINE\[NEWLINEF_\varepsilon(u)=-\sum\limits_{i,j} c^\varepsilon_{i,j}u_iu_jNEWLINE\]NEWLINE is studied. Here, a spin variable \(u\) is defined on a portion of a cubic lattice \(\varepsilon \mathbb Z_d\cap\Omega\), \(\Omega\) being a regular bounded open set, and takes values in \(\{-1,1\}\). For the constants \(c^\varepsilon_{i,j}\) being nonnegative and satisfying suitable coercivity and decay assumptions, it is shown that all possible \( \Gamma \)-limits of surface scalings of the functionals \(F_\varepsilon\) are finite on \(BV(\Omega;\{\pm1\})\) and of the form NEWLINE\[NEWLINE\int\limits_{S_u}\varphi(x,\nu_u)\,d{\mathcal H}^{d-1}.NEWLINE\]NEWLINE If such decay assumptions are violated, it is shown that one may approximate nonlocal functionals of the form NEWLINE\[NEWLINE\int\limits_{S_u}\varphi(\nu_u)\,d{\mathcal H}^{d-1}+ {\int\limits_\Omega} { \int\limits_\Omega} K(x,y) g(u(x),u(y))\,dxdy.NEWLINE\]NEWLINE Two relevant examples: fractional perimeters and Ohta-Kawasaki-type energies are approximated. Eventually, a general criterion for a ferromagnetic behavior of the energies \(F_\varepsilon\) even when the constants \(c^\varepsilon_{i,j}\) change sign is provided. If such a 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 above form.