Journal
PHYSICAL REVIEW E
Volume 79, Issue 4, Pages -Publisher
AMER PHYSICAL SOC
DOI: 10.1103/PhysRevE.79.041108
Keywords
eigenvalues and eigenfunctions; fermion systems; matrix algebra; quantum statistical mechanics
Categories
Funding
- Division Of Mathematical Sciences
- Direct For Mathematical & Physical Scien [0804431] Funding Source: National Science Foundation
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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.
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