The Phase Transition in 5 Point Energy Minimization
Abstract
Let R_s(r)=sign(s)/r^s be the Riesz s-energy potential. (This is the usual power-law potential.) This monograph proves the existence of a computable number S=15.048... such that the triangular bi-pyramid is the unique minimizer with respect to R_s, amongst all 5-point configurations on the sphere, if and only if s lies in (-2,0) or (0,S). This establishes the existence of the long-conjectured phase transition constant in 5-point energy minimization.
Keywords
Cite
@article{arxiv.1610.03303,
title = {The Phase Transition in 5 Point Energy Minimization},
author = {Richard Evan Schwartz},
journal= {arXiv preprint arXiv:1610.03303},
year = {2016}
}
Comments
This is a rigorous computer-assisted proof. Software is available to download. The proof in this version is identical to the proof in the first two versions. In this version I updated the preface and the introduction to reflect improvements I made to the computer code. Also, I fixed typos in equations 15.7-15.9. The final result from this section (Inequality 1) is unchanged