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 $N\to\infty$. 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 $\Gamma$-convergence of such interactions, and describe minimizers on the 1-dimensional flat torus in the absence of external field, for all $N$.