2008/10/28 by Antonello Scardicchio, A. Scardicchio, Chase E. Zachary +3 · 77 citations
Mathematics · Physics and Astronomy · #Combinatorics #Determinantal point process #Dimension (graph theory) #Distribution (mathematics) #Eigenvalues and eigenvectors #Euclidean geometry #Euclidean space #Geometry #Isotropy #Mathematical analysis #Mathematics #Physics #Point process #Point processes and geometric inequalities #Quantum mechanics #Random Matrices and Applications #Random matrix #Saddle point #Statistical physics #Statistics #Stochastic processes and statistical mechanics #Voronoi diagram #cond-mat.stat-mech
paper · pdf · doi:10.1103/physreve.79.041108
published in Physical Review E 79(4), 041108 (American Physical Society) · 42 pages, 17 figures
arxiv created 2008/10/28 · openalex publication_date 2009/04/06 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/05
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 NxN matrices and then extrapolate to N-->infinity . 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 [S. Torquato, A. Scardicchio, and C. E. Zachary, J. Stat. Mech.: Theory Exp. (2008) P11019.], 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.