arXiv subjects

Ryan W. Matzke

Publications and source records attributed to Ryan W. Matzke.

11 recordsLinked to original sources

arXiv2026

Discrepancy of determinantal point processes on compact, connected two-point homogeneous spaces

We study the $L^{\infty}$ discrepancy of point sets generated by determinantal point processes on all compact, connected two-point homogeneous spaces, namely spheres and projective spaces. Using concentration inequalities and variance estimates for the number of points in metric balls, we derive general upper bounds for the discrepancy of homogeneous determinantal point processes. In the particular case of the harmonic ensemble, we show that the discrepancy of $N$ points is $O((N^{1-1/D})^{1/2}\log N)$ with high probability, where $D$ denotes the real dimension of the manifold. For the projective ensemble on $\mathbb{CP}^d$, we obtain the sharper bound $O((N^{1-1/D}\log N)^{1/2})$. These results extend previously known discrepancy estimates for determinantal point processes on the sphere to all compact, connected two-point homogeneous spaces.

math.CA↗

arXiv2026

Energy, Polarization, and Separation of Greedy Sequences for Riesz and Green Kernels

We investigate the asymptotic behavior of greedy $s$-Riesz and Green energy sequences $\{x_{n}\}_{n=1}^{\infty}$ on the unit sphere $\mathbb{S}^{d} \subset \mathbb{R}^{d+1}$, where each point $x_n$ is defined as the minimizer of the discrete potential generated by the preceding points $x_1, x_2, ..., x_{n-1}$. We show that the greedy sequence attains optimal growth behavior for the second-order term of the Green and Riesz $s$-energies when $d-2 \leq s < d$. The main idea is to establish the bounds on polarization using well-separation properties of the greedy configurations.

math.CA↗

arXiv2024

Geodesic Distance Riesz Energy on Projective Spaces

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 Tóth 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} dμ(x) dμ(y) \,\,\, \text{ for } \,\,\, s<d \end{equation} and find ranges of the parameter $s$ for which the energy is minimized by the uniform measure $σ$ on $\mathbb{FP}^d$. To this end, we use various methods of harmonic analysis, such as Cesàro averages of Jacobi expansions and $A_1$ inequalities, and establish a rather general theorem guaranteeing that certain energies with singular kernels are minimized by $σ$. 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).

math.CA↗

arXiv2024

Babai Numbers and Babai Spectra of Paths and Cycles

We study Babai numbers and Babai $k$-spectra of paths and cycles. We completely determine the Babai numbers of paths $P_n$ for $n>1$ and $1 \leq k \leq n-1$, and the Babai $k$-spectra for $P_n$ when $1 \leq k \leq n/2$. We also completely determine Babai numbers and Babai $k$-spectra of all cycles $C_n$ for $k \in \{1,2\}$ and $n \geq 3$ if $k=1$ and $n > 3$ if $k=2$.

math.CO↗

arXiv2024

Riesz Energy with a Radial External Field: When is the Equilibrium Support a Sphere?

We consider Riesz energy problems with radial external fields. We study the question of whether or not the equilibrium is the uniform distribution on a sphere. We develop general necessary as well as general sufficient conditions on the external field that apply to powers of the Euclidean norm as well as certain Lennard--Jones type fields. Additionally, in the former case, we completely characterize the values of the power for which dimension reduction occurs in the sense that the support of the equilibrium measure becomes a sphere. We also briefly discuss the relation between these problems and certain constrained optimization problems. Our approach involves the Frostman characterization, the Funk--Hecke formula, and the calculus of hypergeometric functions.

math.CA↗

arXiv2024

Riesz Energy, $L^2$ Discrepancy, and Optimal Transport of Determinantal Point Processes on the Sphere and the Flat Torus

Determinantal point processes exhibit an inherent repulsive behavior, thus providing examples of very evenly distributed point sets on manifolds. In this paper, we study the so-called harmonic ensemble, defined in terms of Laplace eigenfunctions on the sphere $\mathbb{S}^d$ and the flat torus $\mathbb{T}^d$, and the so-called spherical ensemble on $\mathbb{S}^2$, which originates in random matrix theory. We extend results of Beltrán, Marzo and Ortega-Cerdà on the Riesz $s$-energy of the harmonic ensemble to the nonsingular regime $s<0$, and as a corollary find the expected value of the spherical cap $L^2$ discrepancy via the Stolarsky invariance principle. We find the expected value of the $L^2$ discrepancy with respect to axis-parallel boxes and Euclidean balls of the harmonic ensemble on $\mathbb{T}^d$. We also show that the spherical ensemble and the harmonic ensemble on $\mathbb{S}^2$ and $\mathbb{T}^2$ with $N$ points attain the optimal rate $N^{-1/2}$ in expectation in the Wasserstein metric $W_2$, in contrast to i.i.d. random points, which are known to lose a factor of $(\log N)^{1/2}$.

math.CA↗

arXiv2023

Minimizers for an aggregation model with attractive-repulsive interaction

We solve explicitly a certain minimization problem for probability measures involving an interaction energy that is repulsive at short distances and attractive at large distances. We complement earlier works by showing that part of the remaining parameter regime all minimizers are uniform distributions on a surface of a sphere, thus showing concentration on a lower dimensional set. Our method of proof uses convexity estimates on hypergeometric functions.

math.AP↗

arXiv2023

A random line intersects $\mathbb{S}^2$ in two probabilistically independent locations

We consider random lines in $\mathbb{R}^3$ (random with respect to the kinematic measure) and how they intersect $\mathbb{S}^2$. It is known that the entry point and the exit point behave like \textit{independent} uniformly distributed random variables. We give a new proof using bilinear integral geometry and use this approach to show that this property is extremely rare: if $K \subset \mathbb{R}^n$ is a bounded, convex domain with smooth boundary with this property (i.e., the intersection points with a random line are independent), then $n=3$ and $K$ is a ball.

math.PR↗

arXiv2023

Polarization and Greedy Energy on the Sphere

We investigate the behavior of a greedy sequence on the sphere $\mathbb{S}^d$ defined so that at each step the point that minimizes the Riesz $s$-energy is added to the existing set of points. We show that for $0<s<d$, the greedy sequence achieves optimal second-order behavior for the Riesz $s$-energy (up to constants). In order to obtain this result, we prove that the second-order term of the maximal polarization with Riesz $s$-kernels is of order $N^{s/d}$ in the same range $0<s<d$. Furthermore, using the Stolarsky principle relating the $L^2$-discrepancy of a point set with the pairwise sum of distances (Riesz energy with $s=-1$), we also obtain a simple upper bound on the $L^2$-spherical cap discrepancy of the greedy sequence and give numerical examples that indicate that the true discrepancy is much lower.

math.CA↗

arXiv2023

Optimal Measures for Multivariate Geometric Potentials

We study measures and point configurations optimizing energies based on multivariate potentials. The emphasis is put on potentials defined by geometric characteristics of sets of points, which serve as multi-input generalizations of the well-known Riesz potentials for pairwise interaction. One of such potentials is volume squared of the simplex with vertices at the $k \ge 3$ given points: we show that the arising energy is maximized by balanced isotropic measures, in contrast to the classical two-input energy. These results are used to obtain interesting geometric optimality properties of the regular simplex. As the main machinery, we adapt the semidefinite programming method to this context and establish relevant versions of the $k$-point bounds.

math.CA↗

arXiv2022

Riesz and Green energy on projective spaces

In this paper we study Riesz, Green and logarithmic energy on two-point homogeneous spaces. More precisely we consider the real, the complex, the quaternionic and the Cayley projective spaces. For each of these spaces we provide upper estimates for the mentioned energies using determinantal point processes. Moreover, we determine lower bounds for these energies of the same order of magnitude.

math.CA↗