The Geometry of Optimal and Near-Optimal Riesz Energy Configurations

Abstract

This thesis discusses recent and classical results concerning the asymptotic properties (as N gets large) of ``ground state' configurations of N particles restricted to a compact set A of Hausdorff dimension d interacting through through an inverse power law 1/r^s for some s>0.

It has been observed that, as s becomes large, ground state configurations approach best-packing configurations on A. When d=2, it is generally believed that ground state configurations form a hexagonal lattice. This thesis aims to justify this belief in the case d=2 through the study of geometric inequalities for polygons. Specifically, it is shown that, when s is large, a normalized energy associated to interactions from particles that are ``nearest neighbors' to a fixed point in the configuration is minimized when the nearest neighbors form a regular polygon. This technique provides new lower bounds for the energy for 2-dimensional ground state configurations.