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→∞\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$N\rightarrow \infty $$\end{document}. 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\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ \Gamma $$\end{document}-convergence of such interactions, and describe minimizers on the 1-dimensional flat torus in the absence of external field, for all N.