Nonlocal-interaction vortices

We consider sequences of quadratic non-local functionals, depending on a small parameter $\text{e}$, that approximate the Dirichlet integral by a well-known result by Bourgain, Brezis and Mironescu. Similarly to what is done for hard-core approximations to vortex energies in the case of the Dirichlet integral, we further scale such energies by $\mathrm{\mid }\log \text{e}{\mathrm{\mid }}^{-1}$ and restrict them to $S^1$-valued functions. We introduce a notion of convergence of functions to integral currents with respect to which such energies are equi-coercive, and show the converge to a vortex energy, similarly to the limit behaviour of Ginzburg-Landau energies at the vortex scaling.