Vortices and Magnetization in Kac's Model

We consider a 2-dimensional planar rotator on a large, but finite lattice with a ferromagnetic Kac potential $J_\gamma(i)=\gamma^2J(\gamma i)$, $J$ with compact support. The system is subject to boundary conditions with vorticity. Using a Glauber like dynamics, we compute minimizers of the free energy functional at low temperature, i.e. in the regime of phase transition. We have the numerical evidence of a vortex structure for minimizers, which present many common features with those of the Ginzburg-Landau functional.