Nearest-neighbor Distribution Function Research Articles

Disordered Heterogeneous Universe: Galaxy Distribution and Clustering across Length Scales

The studies of disordered heterogeneous media and galaxy cosmology share a common goal: analyzing the disordered distribution of particles and/or building blocks at microscales to predict physical properties of the medium at macroscales, whether it be a liquid, colloidal suspension, composite material, galaxy cluster, or entire Universe. The theory of disordered heterogeneous media provides an array of theoretical and computational techniques to characterize a wide class of complex material microstructures. In this work, we apply them to describe the disordered distributions of galaxies obtained from recent suites of dark matter simulations. We focus on the determination of lower-order correlation functions, void and particle nearest-neighbor functions, certain cluster statistics, pair-connectedness functions, percolation properties, and a scalar order metric to quantify the degree of order. Compared to analogous homogeneous Poisson and typical disordered systems, the cosmological simulations exhibit enhanced large-scale clustering and longer tails in the void and particle nearest-neighbor functions, due to the presence of quasi-long-range correlations imprinted by early Universe physics, with a minimum particle separation far below the mean nearest-neighbor distance. On large scales, the system appears hyperuniform, as a result of primordial density fluctuations, while on the smallest scales, the system becomes almost antihyperuniform, as evidenced by its number variance. Additionally, via a finite-scaling analysis, we compute the percolation threshold of the galaxy catalogs, finding this to be significantly lower than for Poisson realizations (at reduced density ηc=0.25 in our fiducial analysis compared to ηc=0.34), with strong dependence on the mean density; this is consistent with the observation that the galaxy distribution contains voids of up to 50% larger radius. However, the two sets of simulations appear to share the same fractal dimension on scales much larger than the average intergalaxy separation, implying that they lie in the same universality class. We also show that the distribution of galaxies is a highly correlated disordered system (relative to the uncorrelated Poisson distribution), as measured by the τ order metric. Finally, we consider the ability of large-scale clustering statistics to constrain cosmological parameters, such as the Universe’s expansion rate, using simulation-based inference. Both the nearest-neighbor distribution and pair-connectedness function (which includes contributions from correlation functions of all order) are found to considerably tighten bounds on the amplitude of quantum-mechanical fluctuations from inflation at a level equivalent to observing 25 times more galaxies. The pair-connectedness function in particular provides a useful alternative to the standard three-particle correlation, since it contains similar large-scale information to the three-point function, can be computed highly efficiently, and can be straightforwardly extended to small scales (though likely requires simulation-based modeling). This work provides the first application of such techniques to cosmology, providing both a novel system to test heterogeneous media descriptors and a tranche of new tools for cosmological analyses. A range of extensions are possible, including implementation on observational data; this will require further study on various observational effects, necessitating high-resolution simulations.6 MoreReceived 16 July 2022Revised 2 January 2023Accepted 10 January 2023DOI:https://doi.org/10.1103/PhysRevX.13.011038Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article’s title, journal citation, and DOI.Published by the American Physical SocietyPhysics Subject Headings (PhySH)Research AreasLarge scale structure of the UniverseGravitation, Cosmology & AstrophysicsInterdisciplinary Physics

Nearest neighbor spatial statistics as a framework for analyzing spatial patterns from image data.

Quantitatively describing spatial patterns in biological substrates is a difficult task that requires careful consideration of labeling methods, imaging modality, and the motivating hypothesis. This problem is commonly addressed using intensity correlation paradigms which correlate the brightness between fluorescence images of co-labeled biomolecules as a measure of their colocalization. These approaches ultimately characterize the superposition of fluorescence signals and thus are limited to binary assessment of spatial relationships at the scale of the image resolution. We’ve previously demonstrated an image analysis framework, which leverages nearest neighbor (NN) distribution functions (F, G, and L) for richer, more informative quantification of the spatial distribution of immunofluorescently imaged biomolecules on all spatial scales captured within the image. Briefly, biomolecule signals and relevant biological structures are segmented, biomolecules are localized relative to landmarks, which are either structures or other biomolecules, by measuring NN distances between biomolecule and landmark voxels, and finally their global spatial association is assessed by comparing the resulting distance distribution to a null hypothesis distribution, in our case a homogenous Poisson distribution which represents complete spatial randomness. The sign of the difference between the distributions indicates repulsion or attraction of the biomolecule to the landmark and the magnitude indicates the relationship's strength. Ongoing efforts aim to further validate this framework by characterizing its sensitivity to image segmentation parameters and signal concentration, and the impact on measured spatial relationships of our methods for generating the null distribution and measuring distances within an image of anisotropic resolution. Preliminary results indicate our framework is robust over a wide range of input parameters and data quality, and our custom methods introduce errors that are smaller than the image's precision. Thus, this approach has wide applicability for scientific image analysis.

A foraging ocean sunfish and the ‘nearest neighbor’ problem

A predator that preys on randomly-distributed stationary energetically-equivalent small prey will probably choose its next prey to be the nearest one. But what if no prey is found within the detection range of the predator? It is hypothesized that in this case the predator will move along an arbitrary chosen direction until a prey is detected, and turn towards it. In a stochastic environment this strategy leads to a certain distribution function of distances that the predator moves between consequent prey catches. It is shown that when the detection range of the predator exceeds the average distance between prey, this distribution function becomes the nearest neighbor distribution function, whereas; wherew when the detection range is small as compared with the average distance between prey, it becomes the exponential distribution, as the distribution of distances between neighbors on a line. In the first case, the average distance between catches becomes roughly half the average distance between prey; in the second case, it becomes inversely proportional to the square of the detection range. Ocean sunfish preys on practically stationary jellyfish at depth of more than a hundred meters, in dim light. Plausibly, it can detect jellyfish only at close quarters, and hence its detection range is probably small as compared with the average distance between prey. Analysis of the tracking data from seven animals over a few days yielded many thousands of swimming segments separating consequent prey catches. Indeed, lengths of these segments were shown to have the exponential distribution. This finding not only supports the initial hypothesis of this study, but also reveals the fragility of the energetic balance of this animal. A two-fold decrease in the detection range (e.g. due to a decreased visibility) is expected to increase the average distance it moves between catches four-fold, and hence decrease its specific energy intake (the number of jellyfishes per distance moved) by the same rate.

Topographic Effects on the Spatial Species Associations in Diverse Heterogeneous Tropical Evergreen Forests

Studying spatial patterns and habitat association of plant communities may provide understanding of the ecological mechanisms and processes that maintain species coexistence. To conduct assessments of correlation between community compositions and habitat association, we used data from two topographically different plots with 2 ha area in tropical evergreen forests with the variables recorded via grid systems of 10 × 10 m subplots in Northern-Central Vietnam. First, we tested the relationship between community composition and species diversity indices considering the topographical variables. We then assessed the interspecific interactions of 20 dominant plant species using the nearest-neighbor distribution function, Dij(r), and Ripley’s K-function, Kij(r). Based on the significant spatial association of species pairs, indices of interspecific interaction were calculated by the quantitative amounts of the summary statistics. The results showed that (i) community compositions were significantly influenced by the topographic variables and (ii) almost 50% significant pairs of species interactions were increased with increasing spatial scales up to 10–15 m, then declined and disappeared at scales of 30–40 m. Segregation and partial overlap were the dominant association types and disappeared at larger spatial scales. Spatial segregation, mixing, and partial overlap revealed the important species interactions in maintaining species coexistence under habitat heterogeneity in diverse forest communities.

Statistical functions used for spatial modelling due to assessment of landslide distribution and landscape-interaction factors in Iran

Landslides influence the capacity for safe and sustainable development of mountainous environments. This study explores the spatial distribution of and the interactions between landslides that are mapped using global positioning system (GPS) and extensive field surveys in Mazandaran Province, Iran. Point-pattern assessment is undertaken using several univariate summary statistical functions, including pair correlation, spherical-contact distribution, nearest-neighbor analysis, and O-ring analysis, as well as bivariate summary statistics, and a mark-correlation function. The maximum entropy method was applied to prioritize the factors controlling the incidence of landslides and the landslides susceptibility map. The validation processes were considered for separated 30% data applying the ROC curves, fourfold plot, and Cohen’s kappa index. The results show that pair correlation and O-ring analyses satisfactorily predicted landslides at scales from 1 to 150 ​m. At smaller scales, from 150 to 400 ​m, landslides were randomly distributed. The nearest-neighbor distribution function show that the highest distance to the nearest landslide occurred in the 355 ​m. The spherical-contact distribution revealed that the patterns were random up to a spatial scale of 80 ​m. The bivariate correlation functions revealed that landslides were positively linked to several linear features (including faults, roads, and rivers) at all spatial scales. The mark-correlation function showed that aggregated fields of landslides were positively correlated with measures of land use, lithology, drainage density, plan curvature, and aspect, when the numbers of landslides in the groups were greater than the overall average aggregation. The results of analysis of factor importance have showed that elevation (topography map scale: 1:25,000), distance to roads, and distance to rivers are the most important factors in the occurrence of landslides. The susceptibility model of landslides indicates an excellent accuracy, i.e., the AUC value of landslides was 0.860. The susceptibility map of landslides analyzed has shown that 35% of the area is low susceptible to landslides.

Representative volume elements for the simulation of isotropic composites highly filled with monosized spheres

A method is proposed for generating reliable representative volume elements (RVEs) that allows reducing the statistical analysis required for the simulation of the mechanical behavior of isotropic composites highly filled with monosized spheres. The method combines (i) an algorithm inspired from molecular dynamics and associated with an analytical equation of state, and (ii) a geometrical analysis using the two-point correlation function and a nearest-neighbor distribution function. A restrictive selection process is defined, which leads to microstructures reasonably close to randomness and isotropy. The pertinence of the proposed generation and selection of RVEs is confirmed by the simulation of their elastic behavior with the finite element method. In particular, it is shown how the selection procedure allows reducing the computational effort required to reach reliable elastic moduli by operating on a limited number of suitable RVEs. The results are in good agreement with the generalized self-consistent model and with original experimental data obtained on a composite where an acrylate matrix was reinforced by sifted glass beads.

Intracell Interference Characterization and Cluster Interference for D2D Communication

The homogeneous spatial Poisson point process (SPPP) is widely used for spatial modeling of mobile terminals (MTs). This process is characterized by a homogeneous distribution, complete spatial independence, and constant intensity measure. However, it is intuitive to understand that the locations of MTs are neither homogeneous, due to inhomogeneous terrain, nor independent, due to homophilic relations. Moreover, the intensity is not constant due to mobility. Therefore, assuming an SPPP for spatial modeling is too simplistic, especially for modeling realistic emerging device-centric frameworks such as device-to-device (D2D) communication. In this paper, assuming inhomogeneity, positive spatial correlation, and random intensity measure, we propose a doubly stochastic Poisson process, a generalization of the homogeneous SPPP, to model D2D communication. To this end, we assume a permanental Cox process (PCP) and propose a novel Euler-Characteristic-based approach to approximate the nearest-neighbor distribution function. We also propose a threshold and spatial distances from an excursion set of a chi-square random field as interference control parameters for different cluster sizes. The spatial distance of the clusters is incorporated into a Laplace functional of a PCP to analyze the average coverage probability of a cellular user. A closed-form approximation of the spatial summary statistics is in good agreement with empirical results, and its comparison with an SPPP authenticates the correlation modeling of D2D nodes.

Nearest Neighborhood Characteristics of a Tropical Mixed Broadleaved Forest Stand

Structural complexity and local biodiversity of species-rich tropical forests can be characterized by their spatial patterns, which contribute to species intra- and interspecific interactions. Aiming to describe spatial patterns of species at fine spatial scales, we applied the quantitative analyses based on the relationships of nearest neighbors of conspecific and heterospecific trees. In a two-hectare plot of a tropical broadleaved forest stand in central Vietnam with minimal human influence, all tree individuals with diameter at breast height ≥ 2.5 cm were mapped and their characteristics were recorded. We applied two different types of analyses: (1) Intraspecific structural characteristics using nearest neighbor statistics; (2) overall interspecific associations through a classification scheme based on bivariate nearest neighbor distribution function D12(r) and Ripley’s K function K12(r). The findings showed that: (1) Most of studied species in the forest were highly mixed with other species, while conspecifics were regular to aggregated distribution at small spatial scales. Tree individuals with different diameter values were surrounded by heterospecific trees; (2) The majority of 306 species-species pairs showed spatial independence (66.7%), whereas 29.8% of all species showed an overall positive association and negative association consisted only a small percentage (3.5%) up to spatial scales of 50 m. We found significant evidences of the main ecological theories such as dispersal limitation, Neutral theory, Janzen-Connell hypothesis, and other effects like the stochastic dilution. We suggest using both the bivariate distribution of the structural parameters and the spatial point pattern analysis based on nearest neighbor distance as advantageous approaches for further understanding of population structure, as well as discovering and protecting biodiversity in the future.

Spatial distribution of nuclei in progressive nucleation: Modeling and application

Phase transformations ruled by non-simultaneous nucleation and growth do not lead to random distribution of nuclei. Since nucleation is only allowed in the untransformed portion of space, positions of nuclei are correlated. In this article an analytical approach is presented for computing pair-correlation function of nuclei in progressive nucleation. This quantity is further employed for characterizing the spatial distribution of nuclei through the nearest neighbor distribution function. The modeling is developed for nucleation in 2D space with power growth law and it is applied to describe electrochemical nucleation where correlation effects are significant. Comparison with both computer simulations and experimental data lends support to the model which gives insights into the transition from Poissonian to correlated nearest neighbor probability density.

One-Dimensional Fluids with Second Nearest–Neighbor Interactions

As is well known, one-dimensional systems with interactions restricted to first nearest neighbors admit a full analytically exact statistical-mechanical solution. This is essentially due to the fact that the knowledge of the first nearest-neighbor probability distribution function, $p_1(r)$, is enough to determine the structural and thermodynamic properties of the system. On the other hand, if the interaction between second nearest-neighbor particles is turned on, the analytically exact solution is lost. Not only the knowledge of $p_1(r)$ is not sufficient anymore, but even its determination becomes a complex many-body problem. In this work we systematically explore different approximate solutions for one-dimensional second nearest-neighbor fluid models. We apply those approximations to the square-well and the attractive two-step pair potentials and compare them with Monte Carlo simulations, finding an excellent agreement.

Robust nonparametric quantification of clustering density of molecules in single-molecule localization microscopy.

We report a robust nonparametric descriptor, J′(r), for quantifying the density of clustering molecules in single-molecule localization microscopy. J′(r), based on nearest neighbor distribution functions, does not require any parameter as an input for analyzing point patterns. We show that J′(r) displays a valley shape in the presence of clusters of molecules, and the characteristics of the valley reliably report the clustering features in the data. Most importantly, the position of the J′(r) valley () depends exclusively on the density of clustering molecules (ρc). Therefore, it is ideal for direct estimation of the clustering density of molecules in single-molecule localization microscopy. As an example, this descriptor was applied to estimate the clustering density of ptsG mRNA in E. coli bacteria.

Robust Nonparametric Descriptors for Clustering Quantification in Single-Molecule Localization Microscopy

Single-molecule localization microscopy (SMLM) has been utilized broadly in imaging biological molecules in various biological systems, allowing quantitative analyses on the spatial organizations and patterns of these molecules. However, parameters are needed in many of the currently available methods or algorithms, likely introducing subjective bias in the analyses. Here, we report a robust nonparametric descriptor, J'(r) , for quantifying the spatial organization of molecules in single-molecule localization microscopy. J'(r) , based on nearest neighbor distribution functions, does not require any parameter as an input for analyzing point patterns. We show that it displays a valley shape in the presence of clusters of molecules, and the characteristics of the valley reliably report the clustering features in the data. More importantly, the position of the J'(r) valley (rJ'm) depends exclusively on the density of clustering molecules. Therefore, it is ideal for direct measurements of clustering density of molecules in single-molecule localization microscopy.

Determination of accurate, mean bond lengths from radial distribution functions.

The mean bond length d between a central atom and its nearest neighbors can be estimated from the position of the first peak in the radial distribution function g(r). However, as we demonstrate here, this estimate does not allow one to deduce temperature-induced changes in d. Instead, skewness has to be included into the analysis, which can be achieved, for example, via the skew normal distribution (SND). Fits to the first peak using the SND give bond length in good agreement with direct measurements of nearest-neighbor distribution functions in crystals as well as with a Voronoi-tessellation based detection of nearest-neighbors in liquids. While the location of the first peak in g(r) may shift to smaller values with increasing temperature for three studied liquids-argon, copper, and the bulk-metallic-glass (BMG) forming alloy Zr60Cu30Al10-we find our improved estimates of d to systematically increase with temperature in all cases. Recent conclusions on temperature-induced bond contractions in simple metallic or BMG-forming liquids may therefore have arisen from the neglect of skewness effects.

Instantaneous, parameter-free methods to define a solute's hydration shell.

A range of methods are presented to calculate a solute's hydration shell from computer simulations of dilute solutions of monatomic ions and noble gas atoms. The methods are designed to be parameter-free and instantaneous so as to make them more general, accurate, and consequently applicable to disordered systems. One method is a modified nearest-neighbor method, another considers solute-water Lennard-Jones overlap followed by hydrogen-bond rearrangement, while three methods compare various combinations of water-solute and water-water forces. The methods are tested on a series of monatomic ions and solutes and compared with the values from cutoffs in the radial distribution function, the nearest-neighbor distribution functions, and the strongest-acceptor hydrogen bond definition for anions. The Lennard-Jones overlap method and one of the force-comparison methods are found to give a hydration shell for cations which is in reasonable agreement with that using a cutoff in the radial distribution function. Further modifications would be required, though, to make them capture the neighboring water molecules of noble-gas solutes if these weakly interacting molecules are considered to constitute the hydration shell.

Beyond the single-file fluid limit using transfer matrix method: Exact results for confined parallel hard squares.

We extend the transfer matrix method of one-dimensional hard core fluids placed between confining walls for that case where the particles can pass each other and at most two layers can form. We derive an eigenvalue equation for a quasi-one-dimensional system of hard squares confined between two parallel walls, where the pore width is between σ and 3σ (σ is the side length of the square). The exact equation of state and the nearest neighbor distribution functions show three different structures: a fluid phase with one layer, a fluid phase with two layers, and a solid-like structure where the fluid layers are strongly correlated. The structural transition between differently ordered fluids develops continuously with increasing density, i.e., no thermodynamic phase transition occurs. The high density structure of the system consists of clusters with two layers which are broken with particles staying in the middle of the pore.

A systematic comparison of summary characteristics for quantifying point patterns in ecology

Many functional summary characteristics such as Ripley's K function have been used in ecology to describe the spatial structure of point patterns to aid understanding of the underlying processes. However, their use is poorly guided in ecology because little is understood how well single summary characteristics, or a combination of them, capture the spatial structure of real world patterns. Here, we systematically tested the performance of combinations of eight summary characteristics [i.e. pair correlation function g(r), K‐function K(r), the proportion E(r) of points with no neighbor at distance r, the nearest neighbor distribution function D(r), the spherical contact distribution Hs(r), the kth nearest‐neighbor distribution functions Dk(r), the mean distance nn(k) to the kth neighbor, and the intensity function λ(x)]. To this end we used point pattern data covering a wide range of spatial structures including simulated (stationary) as well as real, possibly non‐stationary, patterns on tree species in a tropical forest in Panamá. To measure the information contained in a given combination of summary characteristics we used simulated annealing to reconstruct the observed patterns based only on the limited information provided by this combination and assessed how well other characteristics of the observed pattern were recovered.We found that the number of summary characteristics required to capture the spatial structure of stationary patterns varied between one (for patterns with near random structures) and three (for patterns with complex cluster and superposition structures), but with a robust ranking g(r), Dk(r), and Hs(r) that was largely independent on pattern idiosyncrasies. Stationary summary characteristics [with ranking g(r), Dk(r), Hs(r), E(r)] captured small‐ to intermediate scale properties of non‐stationary patterns, but for describing large‐scale spatial structures the intensity function was required. Our finding revealed that the current practice in ecology of using only one or two summary characteristics bears danger that essential characteristics of more complex patterns would not be detected. The technique of pattern reconstruction presented here has wide applications in ecology.

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

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.

Application of digital image processing for automatic detection and characterization of cracked constituent particles/inclusions in wrought aluminum alloys

Application of digital image processing is presented for automatic detection and segmentation of cracked iron-rich constituent particles in a hot-rolled wrought Al-alloy where the particle cracks are generated due to monotonic uniaxial tensile loading. Particle cracks (and therefore, cracked particles) with width of 0.5 µm or larger can be detected and segmented. The method is designed for detection of particle cracks generated during tensile testing, and not the particle cracks generated during the thermo-mechanical processing steps such as hot-rolling. The automatic segmentation of cracked iron-rich constituent particles enables robust estimation of second order microstructural descriptors such as nearest neighbor distribution and two-point correlation function of the cracked particles. Such data are very useful to verify and study the theoretical models for damage evolution and fracture in wrought Al-alloys.

Mesoscale structure of diffusion-limited aggregates of colloidal rods and disks

We explore the dependence of the non-universal mesoscale structure of diffusion-limited aggregates upon the shape of their constituent particles. Using random-walker simulations that model anisotropic diffusion in a viscous fluid, we study diffusion-limited aggregation (DLA) of right-circular cylinders having a wide range of length-to-diameter aspect ratios. This single-parameter family of particle shapes allows us to determine the role of particle quasi-dimensionality on the structure of DLA clusters that are composed of effectively one-dimensional thin rods and two-dimensional thin plates. We compare these clusters to those formed by traditional DLA of compact objects by studying the local nature of the interparticle contacts (end–end, end–body, or body–body), the distribution of the number of interparticle contacts, and the wavevector-dependent structure factor S(q) of the resulting clusters. We find that clusters of rods are less dense than those of disks or compact objects of equal volume, yet the long length-scale structure of the DLA clusters conforms to the expected DLA scaling relations. However, the local structure at the particle scale, including the nearest-neighbor distribution functions and dominant collision types, depend strongly on the particle's quasi-dimensionality. We explain the non-universal local structure by introducing the concept of a diffusing particle's ‘touch space’, which incorporates both the particle’s geometry and its anisotropic diffusion.

Point processes in arbitrary dimension from fermionic gases, random matrix theory, andnumber theory

It is well known that one can map certain properties of random matrices, fermionic gases,and zeros of the Riemann zeta function to a unique point process on the real line . Here we analytically provide exact generalizations of such a point process ind-dimensional Euclidean space for any d, which are special cases of determinantal processes. In particular, we obtain then-particle correlationfunctions for any n, which completely specify the point processes in . We also demonstrate that spin-polarized fermionic systems in have these same n-particle correlation functions in each dimension. The point processes for anyd are shown to be hyperuniform, i.e., infinite wavelength densityfluctuations vanish, and the structure factor (or power spectrum)S(k) has a non-analytic behavior at the origin given byS(k)∼|k| (). The latter result implies that the pair correlation functiong2(r) tends to unity for large pair distances with a decay rate that is controlled by the power law1/rd+1, which is a well-known property of bosonic ground states and more recently has beenshown to characterize maximally random jammed sphere packings. We graphically displayone-and two-dimensional realizations of the point processes in order to vividlyreveal their ‘repulsive’ nature. Indeed, we show that the point processes can becharacterized by an effective ‘hard core’ diameter that grows like the square root ofd. The nearest-neighbor distribution functions for these point processes arealso evaluated and rigorously bounded. Among other results, this analysisreveals that the probability of finding a large spherical cavity of radiusr indimension d behaves like a Poisson point process but in dimensiond+1, i.e., this probabilityis given by exp[−κ(d)rd+1] for large r andfinite d, whereκ(d) is a positived-dependent constant.We also show that as d increases, the point process behaves effectively like a spherepacking with a coverage fraction of space that is no denser than1/2d. This coverage fraction has a special significance in the study of sphere packings inhigh-dimensional Euclidean spaces.