  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 ${\mathcal L}={\mathbb
Z}^d$.
