Geodesic Distance Riesz Energy on Projective Spaces

TL;DR

This paper investigates probability measures minimizing Riesz energy based on geodesic distance on projective spaces, revealing phase transitions in minimizers unlike typical cases, using harmonic analysis techniques.

Contribution

It introduces new results on energy minimization on projective spaces, employing harmonic analysis and uncovering multiple phase transitions in minimizers.

Findings

Minimizers are often the uniform measure on projective spaces.

Multiple phase transitions occur in the minimizers of the energy.

Numerical evidence supports the theoretical findings.

Abstract

We study probability measures that minimize the Riesz energy with respect to the geodesic distance $\vartheta (x,y)$ on projective spaces $\mathbb{FP}^d$ (such energies arise from the 1959 conjecture of Fejes T\'oth about sums of non-obtuse angles), i.e. the integral \begin{equation} \frac{1}{s} \int_{\mathbb{FP}^d} \int_{\mathbb{FP}^d} \big( \vartheta (x,y) \big)^{-s} d\mu(x) d\mu (y) \,\,\, \text{ for } \,\,\, s<d \end{equation} and find ranges of the parameter $s$ for which the energy is minimized by the uniform measure $\sigma$ on $\mathbb{FP}^d$. To this end, we use various methods of harmonic analysis, such as Ces\`aro averages of Jacobi expansions and $A_1$ inequalities, and establish a rather general theorem guaranteeing that certain energies with singular kernels are minimized by $\sigma$. In addition, we obtain further results and present numerical evidence, which uncover a peculiar effect that minimizers this energy undergo numerous phase transitions, in sharp contrast with many analogous known examples (even the seemingly similar geodesic Riesz energy on the sphere), which usually have only one transition (between uniform and discrete minimizers).