Asymptotics of k-nearest Neighbor Riesz Energies

We obtain new asymptotic results about systems of N particles governed by Riesz interactions involving k-nearest neighbors of each particle as N ? 8. These results include a generalization to weighted Riesz potentials with external field. Such interactions offer an appealing alternative to other approaches for reducing the computational complexity of an N-body interaction. We find the first-order term of the large N asymptotics and characterize the limiting distribution of the minimizers. We also obtain results about the G-convergence of such interactions, and describe minimizers on the 1-dimensional flat torus in the absence of external field, for all N.