*Preprint*

**Inserted:** 1 oct 2019

**Last Updated:** 1 oct 2019

**Year:** 2019

**Abstract:**

We consider energies on a periodic set ${\mathcal L}$ of ${\mathbb R}^d$ of the form
$\sum_{i,j\in{\mathcal L}} a^\varepsilon_{ij}\

u_i-u_j\

$, defined on spin functions $u_i\in\{0,1\}$, and we suppose that the typical range of the interactions is $R_\varepsilon$ with $R_\varepsilon\to +\infty$, i.e., if $\

i-j\

\le R_\varepsilon$ then $a^\varepsilon_{ij}\ge c>0$. In a discrete-to-continuum analysis, we prove that the overall behaviour as $\varepsilon\to 0$ of such functionals is that of an interfacial energy. The proof is performed using a coarse-graining procedure which associates to scaled functions defined on $\varepsilon{\mathcal L}$ with equibounded energy a family of sets with equibounded perimeter. This agrees with the case of equibounded $R_\varepsilon$ and can be seen as an extension of coerciveness result for short-range interactions, but is different from that of other long-range interaction energies, whose limit exits the class of surface energies. A computation of the limit energy is performed in the case ${\mathcal L}={\mathbb Z}^d$.

**Download:**