TY - JOUR

T1 - Statistical properties of determinantal point processes in high-dimensional Euclidean spaces

AU - Scardicchio, Antonello

AU - Zachary, Chase E.

AU - Torquato, Salvatore

PY - 2009/4/1

Y1 - 2009/4/1

N2 - 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×N matrices and then extrapolate to N→. 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 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=1-4. This homogeneous, isotropic determinantal point process, discussed also in a companion paper, is the high-dimensional generalization of the distribution of eigenvalues on the unit circle of a random matrix from the circular unitary ensemble. 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 we discuss these properties as the dimension d is varied. The results in this paper accompany and complement analytical properties of higher-dimensional determinantal point processes developed in a prior paper.

AB - 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×N matrices and then extrapolate to N→. 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 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=1-4. This homogeneous, isotropic determinantal point process, discussed also in a companion paper, is the high-dimensional generalization of the distribution of eigenvalues on the unit circle of a random matrix from the circular unitary ensemble. 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 we discuss these properties as the dimension d is varied. The results in this paper accompany and complement analytical properties of higher-dimensional determinantal point processes developed in a prior paper.

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U2 - 10.1103/PhysRevE.79.041108

DO - 10.1103/PhysRevE.79.041108

M3 - Article

C2 - 19518174

AN - SCOPUS:65449121536

SN - 1539-3755

VL - 79

JO - Physical Review E - Statistical, Nonlinear, and Soft Matter Physics

JF - Physical Review E - Statistical, Nonlinear, and Soft Matter Physics

IS - 4

M1 - 041108

ER -