The Phase Transition in 5 Point Energy Minimization

TL;DR

This paper proves the existence of a specific phase transition point in 5-point energy minimization on the sphere, identifying the unique minimizer as a triangular bi-pyramid within certain parameter ranges.

Contribution

It establishes the exact value of the phase transition constant for 5-point configurations on the sphere and characterizes the unique minimizer depending on the parameter s.

Findings

Existence of a computable phase transition constant S=15.048...

Triangular bi-pyramid is the unique minimizer for s in (-2,0) or (0,S)

Confirms the long-conjectured 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.