Particle approximation of nonlocal interaction energies

We consider Riesz-type nonlocal energies with general interaction kernels and their discretizations related to particle systems. We prove that the discretized energies $\mathrm{\Gamma }$-converge in the weak-$\ast$ topology to the Riesz functional defined over the space of probability measures. We also address the minimization problem for the discretized energies, and prove the existence of minimal configurations of particles in a very general and natural setting.