Nonlocal-to-local variational limits of energies on partitions

We study, through a $\mathrm{\Gamma }$-convergence procedure, the asymptotic behaviour of a general class of nonlocal energies defined on functions taking values in a finite set $Y=\{y_1,\dots ,y_N\}\subset {\mathbb{R}}^m$. These energies are written in terms of double integrals accounting for pairwise interactions in an open set $\mathrm{\Omega }$, weighted by a kernel $K_{\epsilon }$. If the latter is controlled from above and below by suitable non-negative radial kernels with unit mass concentrating at the origin, then, up to subsequences, the $\mathrm{\Gamma }$-limit has domain $BV(\mathrm{\Omega };Y)$ and is given by a local surface energy. We show that the locality of the $\mathrm{\Gamma }$-limit is not guaranteed if the concentration of mass fails. For a class of reference energies, we obtain a more explicit representation of the limit energy, also for non-radial kernels, recovering the limits of isotropic and anisotropic fractional perimeters. We also prove a homogenization result for periodic kernels.