Asymptotics of $k$-nearest neighbor Riesz energies

TL;DR

This paper derives new asymptotic formulas for systems of particles interacting via k-nearest neighbor Riesz potentials, including weighted cases with external fields, and characterizes their minimizers and limiting distributions.

Contribution

It generalizes asymptotic analysis of Riesz energies to k-nearest neighbor interactions, including weighted potentials and external fields, and studies minimizers and Gamma-convergence.

Findings

First-order asymptotics of k-nearest neighbor Riesz energies

Characterization of limiting distributions of minimizers

Results on Gamma-convergence and minimizers on the 1D torus

Abstract

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 $.