Harmonic Ensemble DPP

• Harmonic ensemble is a determinantal point process defined by projecting onto the first Laplace–Beltrami eigenfunctions, ensuring uniform coverage and strong repulsion.
• It achieves optimal Wasserstein distance rates of N^(-1/d) for dimensions d ≥ 3 and adapts with logarithmic corrections for d = 2, confirming its efficiency in uniform sampling.
• Its construction via constant diagonal kernels and explicit Fourier or lattice analysis underpins applications in numerical integration, mesh generation, and statistical physics.

The harmonic ensemble is a determinantal point process (DPP) constructed on a compact Riemannian manifold, typically defined via projection onto the span of the first N Laplace–Beltrami eigenfunctions, with a special focus on homogeneous and two-point homogeneous manifolds. The kernel of this DPP has constant diagonal (K(x, x) = N), and its principal feature is strong repulsion among points, producing highly uniform spatial distributions. The key quantitative tool for measuring the quality of equidistribution is the expected Wasserstein distance (typically the 2-Wasserstein metric, W₂) between the empirical measure of the random point process and the uniform (volume) probability measure. The harmonic ensemble thus forms a prototypical model for studying the random yet optimally uniform sampling of manifolds, with implications spanning numerical analysis, probability, and mathematical physics.

1. Construction and Characteristics of the Harmonic Ensemble

The harmonic ensemble is defined as the DPP associated with the orthogonal projection kernel onto a finite-dimensional subspace of L²(M), where M is a compact Riemannian manifold (e.g., the sphere Sᵈ or torus 𝕋ᵈ). If {f₁, ..., f_N} is an orthonormal basis (typically Laplacian eigenfunctions), the kernel takes the form

$$K_N(x, y) = \sum_{n=1}^{N} f_n(x) \overline{f_n(y)},$$

and the DPP is the probability measure on N-point configurations {x₁, ..., x_N} (in M) with joint density proportional to det[K_N(x_i, x_j)]_{i, j=1}^{N}. A defining property for the rate results is the constant diagonal: K_N(x, x) ≡ N, ensuring that the "first intensity" (the expected empirical measure) is uniform.

In the context of homogeneous manifolds (those with transitive isometry group), such as spheres, projective spaces, and tori, the harmonic ensemble provides highly symmetric and well-distributed point patterns. Two-point homogeneous manifolds (where the isometry group acts transitively on the set of pairs of points with equal distance) include the classical spheres and their relatives, and in all these cases, the structural properties of the Laplace spectrum transfer to the DPP.

2. Quantitative Equidistribution: Wasserstein Rates

A central quantitative object in assessing the "uniformity" of the harmonic ensemble is the expected 2-Wasserstein distance W₂ between the empirical measure,

$\mu_N = \frac{1}{N} \sum_{n=1}^{N} \delta_{x_n},$

and the Riemannian volume measure Vol normalized as a probability measure. The paper establishes that, for the harmonic ensemble with constant diagonal kernel:

• On homogeneous manifolds of dimension $d \geq 3$,

$$\mathbb{E}[W_2(\mu_N, \mathrm{Vol})] \lesssim N^{-1/d}.$$

• For dimension $d = 2$, there is a logarithmic correction:

$$\mathbb{E}[W_2(\mu_N, \mathrm{Vol})] \lesssim \sqrt{\log N}\, N^{-1/2}.$$

• On two-point homogeneous manifolds (including $d = 2$), the optimal scaling

$$\mathbb{E}[W_2(\mu_N, \mathrm{Vol})] \lesssim N^{-1/d}$$

is attained.

These rates are optimal under general geometric and probabilistic constraints—matching known lower bounds for optimal $N$-point sets and surpassing i.i.d. random point sets, which exhibit an unavoidable logarithmic loss in $d=2$.

3. Variations: Torus Ensembles and Comparison to Other Processes

On the torus 𝕋ᵈ, the harmonic ensemble is constructed via the projection onto lattice Fourier modes,

$$f_j(x) = e^{2\pi i \langle j, x \rangle}, \quad j \in \mathbb{Z}^d, \quad \|j\|_p \leq L,$$

with $N \sim L^d$. The process again has constant first intensity, and the main result (for $d\geq 2$),

$$\mathbb{E}[W_2(\mu_N, \mathrm{Vol})] \lesssim N^{-1/d},$$

is obtained via explicit variance calculations of the Fourier coefficients and application of smoothing inequalities (e.g., Berry–Esseen analogues for Wasserstein metrics). For $d = 2$, explicit lattice counting leads to

$$\mathbb{E}[W_2(\mu_N, \mathrm{Vol})] \lesssim N^{-1/2},$$

up to a logarithmic factor.

The methodology generalizes to other "repulsive" point processes:

• The spherical ensemble (eigenvalues of certain random matrices projected to $S^2$): optimal rate in $d=2$.
• Zeros of Gaussian analytic functions (GAFs) on the sphere: same optimal rate.

These results contrast with i.i.d. sampling, where for $d=2$, only a rate of order $\sqrt{\log N}/N^{1/2}$ is attainable. The harmonic ensemble and its relatives demonstrate how determinantal structure translates into stronger spatial uniformity.

4. Theoretical Mechanisms Underpinning Optimality

The harmonic ensemble's equidistribution properties stem fundamentally from the strong short-range repulsion inherent to determinantal point processes with projection kernels. The constant diagonal property ensures that the mean empirical measure is already perfectly uniform. The key technical ingredient in proving the upper bounds is the variance calculation for the empirical measure integrated against a smooth test function $f$:

$$\mathrm{Var}\left( \sum_{n=1}^N f(x_n) \right) \lesssim N.$$

The smoothing inequality then transfers $L^2$ control to $W_2$ control, yielding convergence at the optimal rate $N^{-1/d}$ for $d\geq 3$. In $d=2$, the variance saturates the lower bound, modulo logarithmic terms.

In two-point homogeneous settings, the explicit algebraic structure of reproducing kernels (e.g., spherical harmonics) allows for direct computation. On the torus, lattice point counting arguments and explicit knowledge of the Fourier spectrum provide the corresponding variance bounds.

5. Robustness and Applicability

The optimal equidistribution property of the harmonic ensemble is robust across geometries: as long as the kernel is a finite-rank projection with constant diagonal, and the underlying manifold supports sufficient harmonic analysis (homogeneous structure), the N^{–1/d} scaling is achieved. This robustness is particularly important for practical applications, such as:

• Numerical integration (quasi-Monte Carlo): The harmonic ensemble DPP provides point sets with minimal discrepancy in Wasserstein distance, a key figure of merit for integration or sampling tasks on manifolds.
• Sampling and mesh generation: In tasks requiring uniform coverage of a manifold, the harmonic ensemble surpasses i.i.d. sampling both theoretically and empirically.
• Random matrix theory and mathematical physics: The distributional and repulsive properties of the harmonic ensemble model phenomena from electron correlations to eigenvalue distributions in random operators.

The analysis extends to DPPs with kernel truncation based on other basis sets (e.g., polyanalytic functions on complex domains or GAF zeros), provided the projection kernel and underlying geometry exhibit comparable regularity.

6. Comparative Table: Expected Wasserstein Distance Rates

Ensemble & Manifold	Dimension d	Expected Rate	Diagonal Kernel Constancy
Harmonic ensemble (DPP)	$d \geq 3$	$N^{-1/d}$	Yes
Harmonic ensemble (DPP)	$d = 2$	$N^{-1/2}$ (log corr.)	Yes
Spherical ensemble	$d = 2$	$N^{-1/2}$	Yes
Zeros of GAF (sphere)	$d = 2$	$N^{-1/2}$	Non-DPP but nearly const.
IID random points	$d \geq 3$	$N^{-1/d}$	No
IID random points	$d = 2$	$\sqrt{\log N}\, N^{-1/2}$	No

This table synthesizes the scaling behaviors and structural prerequisites for achieving optimal equidistribution, highlighting the advantage of DPP mechanisms with constant diagonal kernels.

7. Broader Implications

The asymptotic optimality in Wasserstein equidistribution of the harmonic ensemble demonstrates that determinantal repulsion can be leveraged for efficient and highly uniform sampling strategies on manifolds. Such processes, rooted in the spectral properties of the Laplacian or related operators, offer a valuable interface between spectral geometry, probability, and computational methods. They serve both to advance theoretical understanding of spatial processes on compact spaces and to provide direct practical tools in computation, simulation, and statistical physics. These principles are portable to a wide class of compact geometries where Laplacian eigenfunctions or their analogues are available, so long as constant diagonal projections are maintained (Arias, 27 May 2024).