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→∞ . 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 Γ -convergence of such interactions, and describe minimizers on the 1-dimensional flat torus in the absence of external field, for all N .