Minimal Riesz and logarithmic energies on the Grassmannian $\operatorname{Gr}_{2,4}$

TL;DR

This paper investigates Riesz and logarithmic energies on the Grassmannian $ ext{Gr}_{2,4}$, proving the uniform measure minimizes these energies and providing asymptotic bounds for discrete energies, with explicit constants from a determinantal point process.

Contribution

It establishes the unique minimization of continuous energies by the uniform measure and introduces a determinantal process to compute explicit energy bounds.

Findings

Uniform measure uniquely minimizes continuous energies.

Asymptotic bounds for minimal discrete energies are derived.

Explicit constants obtained via a determinantal point process.

Abstract

We study the Riesz and logarithmic energies on the Grassmannian $\operatorname{Gr}_{2,4}$ of $2$-dimensional subspaces of $\mathbb{R}^4$. We prove that the continuous Riesz and logarithmic energies are uniquely minimized by the uniform measure, and we obtain asymptotic upper and lower bounds for the minimal discrete energies, with matching orders for the next-order terms. Additionally, we define a determinantal point process on $\operatorname{Gr}_{2,4}$ and compute the expected energy of the points coming from this random process, thereby obtaining explicit constants in the upper bounds for the Riesz and logarithmic energies.