Statistical properties of determinantal point processes in high-dimensional Euclidean spaces

TL;DR

This paper analyzes the statistical properties of high-dimensional determinantal point processes, focusing on nearest-neighbor distributions, and introduces numerical methods to evaluate these properties in Euclidean spaces of dimensions 1 to 4.

Contribution

It provides a numerical framework for evaluating statistical functions of determinantal point processes in high dimensions and characterizes the Fermi-sphere point process across multiple dimensions.

Findings

Determinantal point processes' nearest-neighbor distributions can be expressed as determinants and extrapolated to infinite size.

Numerical algorithms enable efficient evaluation of these distributions in Euclidean spaces up to 4 dimensions.

The Fermi-sphere point process generalizes eigenvalue distributions from random matrix theory to higher dimensions.

Abstract

The goal of this paper is to quantitatively describe some statistical properties of higher-dimensional determinantal point processes with a primary focus on the nearest-neighbor distribution functions. Toward this end, we express these functions as determinants of $N\times N$ matrices and then extrapolate to $N\to\infty$. This formulation allows for a quick and accurate numerical evaluation of these quantities for point processes in Euclidean spaces of dimension $d$. We also implement an algorithm due to Hough \emph{et. al.} \cite{hough2006dpa} for generating configurations of determinantal point processes in arbitrary Euclidean spaces, and we utilize this algorithm in conjunction with the aforementioned numerical results to characterize the statistical properties of what we call the Fermi-sphere point process for $d = 1$ to 4. This homogeneous, isotropic determinantal point process, discussed also in a companion paper \cite{ToScZa08}, is the high-dimensional generalization of the distribution of eigenvalues on the unit circle of a random matrix from the circular unitary ensemble (CUE). In addition to the nearest-neighbor probability distribution, we are able to calculate Voronoi cells and nearest-neighbor extrema statistics for the Fermi-sphere point process and discuss these as the dimension $d$ is varied. The results in this paper accompany and complement analytical properties of higher-dimensional determinantal point processes developed in \cite{ToScZa08}.