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

TL;DR

The paper analyzes Riesz and logarithmic energies on the Grassmannian $\operatorname{Gr}_{2,4}$, proving that continuous energies are uniquely minimized by the uniform measure $\sigma$ and deriving precise asymptotics for minimal discrete energies. It develops a determinantal point process on $\operatorname{Gr}_{2,4}$, termed the harmonic ensemble, whose kernel is given in closed form by Gegenbauer polynomials, and uses it to obtain explicit constants for the next-order terms in energy estimates. The authors combine a linear-programming lower-bound framework with jittered sampling upper-bound constructions to establish tight asymptotics across regimes $0<s<4$ and $s=\log$, including hypersingular behavior at $s=4$. They also compute the expected energies of the harmonic ensemble, obtaining explicit constants in the $N^2$ and $N^{1+s/4}$ terms (and the $N^2\log N$ term for $s=4$), and confirm that these random constructions achieve the correct leading-order behavior for minimal energies. Overall, the work provides sharp, explicit asymptotics for energy minimization on the Grassmannian and demonstrates the effectiveness of DPPs in high-dimensional geometric sampling problems with two principal angles.

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.

Figures (3)

• Figure 1: Regions of integration inside the rectangle $R=[-1,1]\times[0,1]$.
• Figure 2: Regions of integration inside the rectangle $[0,\pi]\times[0,\pi/2]$.
• Figure 3: Subdivision of the rectangle $\tilde{R}=[0,\pi]\times[0,\pi/2]$ used in the proofs of \ref{['lemma:integral-dpp-hyper-1', 'lemma:integral-dpp-hyper-2', 'lemma:integral-dpp-hyper-3']}.

Theorems & Definitions (87)

• Proposition 1.1: Bachoc2006
• Proposition 1.2: James1954; Davis1999b
• Corollary 1.3
• Lemma 1.4
• Theorem 1.5: see BorodachovHardinSaff2019
• Proposition 1.6
• Remark 1.7
• Theorem 2.1
• Theorem 2.2
• Theorem 2.3