We investigate the hyperuniformity of marked Gibbs point processes that have weak dependencies among distant points whilst the interactions of close points are kept arbitrary. Various stability and range assumptions are imposed on the Papangelou intensity in order to prove that the resulting point process is not hyperuniform. The scope of our results covers many frequently used models, including Gibbs point processes with a superstable, lower-regular, integrable pair potential, as well as the Widom–Rowlinson model with random radii and Gibbs point processes with interactions based on Voronoi tessellations and nearest-neighbour graphs.

We study the measure of a typical cell in a Voronoi tessellation defined by n independent random points X1, . . ., Xn drawn from an absolutely continuous probability measure μ with density f in ℝd. We prove that the asymptotic distribution of the measure – with respect to dμ = f(x)dx – of the cell containing X1 given X1 = x is independent of x and the density f. We determine all moments of the asymptotic distribution and show that the distribution becomes more concentrated as d becomes large. In particular, we show that the variance converges to 0 exponentially fast in d. We also obtain a bound independent of the density for the rate of convergence of the diameter of a typical Voronoi cell.

We consider the following stochastic model for a mobile service scenario. Consider a stationary Poisson process in Rd, with its points radially ordered with respect to the origin (the anchor); if d = 2, the points may correspond to locations of, e.g. restaurants. A user, with a location different from the origin, asks for the location of the first Poisson point and keeps asking for the location of the next Poisson point until the first time that he/she can be completely certain that he/she knows which Poisson point is his/her nearest neighbour. This waiting time is the communication cost, while the inferred privacy region is a random set obtained by an adversary who only knows the anchor and the points received from the server, where the adversary ‘does the best’ to infer the possible locations of the user. Probabilistic results related to the communication cost and the inferred privacy region are established for any dimension d ≥ 1. Furthermore, special results when d = 1 and particularly when d = 2 are derived.

For a typical cell of a homogeneous Poisson-Voronoi tessellation in ℝd, it is shown that the variance of the volume of the intersection of the typical cell with any measurable subset of ℝd is bounded by the variance of the volume of the typical cell. It is also shown that the variance of the volume of the intersection of the typical cell with a translation of itself is bounded by four times the variance of the volume of the typical cell. These bounds are applied to show large-dimensional volume degeneracy as d tends to ∞. An extension to the kth nearest-point Poisson-Voronoi tessellation for k ≥ 2 is also considered.

In this article a novel model framework to simulate cells and their internal structure is described. The model is agent-based and suitable to simulate single cells with a detailed internal structure as well as multi-cellular compounds. Cells are simulated as a set of many interacting particles, with neighborhood relations defined via a Delaunay triangulation. The interacting sub-particles of a cell can assume specific roles – i.e., membrane sub-particle, internal sub-particle, organelles, etc –, distinguished by specific interaction potentials and, eventually, also by the use of modified interaction criteria. For example, membrane sub-particles may interact only on a two-dimensional surface embedded on three-dimensional space, described via a restricted Delaunay triangulation. The model can be used not only to study cell shape and movement, but also has the potential to investigate the coupling between internal space-resolved movement of molecules and determined cell behaviors.

We consider a stationary Poisson process X of k-flats in ℝd with intensity measure Θ and a measurable set S of k-flats depending on F1,…,Fn∈ X, x∈ℝd, and X in a specific equivariant way. If (F1,…,Fn,x) is properly sampled (in a ‘typical way’) then Θ(S) has a gamma distribution. This result generalizes and unifies earlier work by Miles (1971), Møller and Zuyev (1996), and Zuyev (1999). As a new example, we will show that the volume of the fundamental region of a typical j-face of a stationary Poisson–Voronoi tessellation is conditionally gamma distributed. This is true in the area-biased and the area-debiased cases. In the first case the shape parameter is not integer valued. As another new example, we will show that the generalized integral-geometric contents of the (area-biased and area-debiased) typical j-face of a Poisson hyperplane tessellation are conditionally gamma distributed. In the isotropic case the contents boil down to the mean breadth of the face.

In this paper we consider a tessellation V generated by a homogeneous Poisson process Φ in Rd and, furthermore, the random set of spheres with centres being the points in Φ and having radii equal to half the distance to their closest neighbouring point in Φ. In Rd we give an integral formula for the correlation between the volume of the typical cell and the volume of the sphere in the typical cell, and we also show that this correlation is strictly positive. Furthermore, on the real line we give an analytical expression for the correlation, and in the plane and in space we give simplified integral formulae. Numerical values for the correlation for d = 2,…,7 are also given.

We introduce a new class of spatial-temporal point processes based on Voronoi tessellations. At each step of such a process, a point is chosen at random according to a distribution determined by the associated Voronoi cells. The point is then removed, and a new random point is added to the configuration. The dynamics are simple and intuitive and could be applied to modelling natural phenomena. We prove ergodicity of these processes under wide conditions.

We consider the Voronoi tessellation based on a stationary Poisson process N in ℝd. We provide a complete and explicit description of the Palm distribution describing N as seen from a randomly chosen (typical) point on a k-face of the tessellation. In particular, we compute the joint distribution of the d−k+1 neighbours of the k-face containing the typical point. Using this result as well as a fundamental general relationship between Palm probabilities, we then derive some properties of the typical k-face and its neighbours. Generalizing recent results of Muche (2005), we finally provide the joint distribution of the typical edge (typical 1-face) and its neighbours.

3-D Voronoi's tessellation method has been applied for identifying groups of galaxies in the structure of the Local Supercluster and evaluation of its properties.

We consider the Voronoi tessellation of Euclidean space that is generated by an inhomogeneous Poisson point process whose intensity takes different constant values on sets of some finite partition of the space. Considering the Voronoi cells as marks associated with points of the point process, we prove that the intensity measure (mean measure) of the marked Poisson point process admits an approximate decomposition formula. The true value is approximated by a mixture of respective intensity measures for homogeneous models, while the explicit upper bound for the remainder term can be computed numerically for a large class of practical examples. By the Campbell formula, analogous approximate decompositions are deduced for the Palm distributions of individual cells. This approach makes possible the analysis of a wide class of inhomogeneous-Poisson Voronoi tessellations, by means of formulae and estimates already established for homogeneous cases. Our analysis applies also to the Poisson process modulated by an independent stationary random partition, in which case the error of the approximation of the double-stochastic-Poisson Voronoi tessellation depends on some integrated linear contact distribution functions of the boundaries of the partition elements.

A known gamma-type result for the Poisson process states that certain domains defined through configuration of the points (or ‘particles’) of the process have volumes which are gamma distributed. By proving the corresponding sequential gamma-type result, we show that in some cases such a domain allows for decomposition into subdomains each having independent exponentially distributed volumes. We consider other examples—based on the Voronoi and Delaunay tessellations—where a natural decomposition does not produce subdomains with exponentially distributed volumes. A simple algorithm for the construction of a typical Voronoi flower arises in this work. In our theoretical development, we generalize the classical theorem of Slivnyak, relating it to the strong Markov property of the Poisson process and to a result of Mecke and Muche (1995). This new theorem has interest beyond the specific problems being considered here.

We define a class of tessellation models based on perturbing or deforming standard tessellations such as the Voronoi tessellation. We show how distributions over this class of ‘deformed’ tessellations can be used to define prior distributions for models based on tessellations, and how inference for such models can be carried out using Markov chain Monte Carlo methods; stability properties of the algorithms are investigated. Our approach applies not only to fixed dimension problems, but also to variable dimension problems, in which the number of cells in the tessellation is unknown. We illustrate our methods with two real examples. The first relates to reconstructing animal territories, represented by the individual cells of a tessellation, from observation of an inhomogeneous Poisson point process. The second example involves the analysis of an image of a cross-section through a sample of metal, with the tessellation modelling the micro-crystalline structure of the metal.

We define and analyse a random coverage process of the d-dimensional Euclidean space which allows us to describe a continuous spectrum that ranges from the Boolean model to the Poisson–Voronoi tessellation to the Johnson–Mehl model. As for the Boolean model, the minimal stochastic setting consists of a Poisson point process on this Euclidean space and a sequence of real valued random variables considered as marks of this point process. In this coverage process, the cell attached to a point is defined as the region of the space where the effect of the mark of this point exceeds an affine function of the cumulative effect of all marks. This cumulative effect is defined as the shot-noise process associated with the marked point process. In addition to analysing and visualizing this spectrum, we study various basic properties of the coverage process such as the probability that a point or a pair of points be covered by a typical cell. We also determine the distribution of the number of cells which cover a given point, and show how to provide deterministic bounds on this number. Finally, we also analyse convergence properties of the coverage process using the framework of closed sets, and its differentiability properties using perturbation analysis. Our results require a pathwise continuity property for the shot-noise process for which we provide sufficient conditions. The model in question stems from wireless communications where several antennas share the same (or different but interfering) channel(s). In this case, the area where the signal of a given antenna can be received is the area where the signal to interference ratio is large enough. We describe this class of problems in detail in the paper. The results obtained allow us to compute quantities of practical interest within this setting: for instance the outage probability is obtained as the complement of the volume fraction; the law of the number of cells covering a point allows us to characterize handover strategies, and so on.

We study Poisson-Voronoi tessellations in three-dimensional hyperbolic spaces, and give explicit expressions for mean surface area, mean perimeter length, and mean number of vertices of their cells. Furthermore we compare these mean characteristics with those for Poisson-Voronoi tessellations in three-dimensional Euclidean spaces. It is shown that, as the absolute value of the curvature of hyperbolic spaces increases from zero to infinity, these mean characteristics increase monotonically from those for the Euclidean case to infinity.

Consider the Delaunay graph and the Voronoi tessellation constructed with respect to a Poisson point process. The sequence of nuclei of the Voronoi cells that are crossed by a line defines a path on the Delaunay graph. We show that the evolution of this path is governed by a Markov chain. We study the ergodic properties of the chain and find its stationary distribution. As a corollary, we obtain the ratio of the mean path length to the Euclidean distance between the end points, and hence a bound for the mean asymptotic length of the shortest path.

We apply these results to define a family of simple incremental algorithms for constructing short paths on the Delaunay graph and discuss potential applications to routeing in mobile communication networks.

There were three objectives to the work undertaken for this paper: (1) to provide a comprehensive characterization of the statistical properties of arrays of β-cell somata; (2) to develop a model that simulates cellular arrays with the same properties; and (3) to use this model to examine whether the array of β-cells should be viewed as one array or as two arrays, one each for its OFF- and ON-center cells. β-cells are morphological correlates of the electrophysiological X-cells and those β-cells whose dendrites stratify within the outer and inner sublamina of the retina's inner plexiform layer correspond, respectively, to OFF- and ON-center X-cells. Arrays of peripheral β-cell somata from two retinas were studied. A Delaunay triangulation and a Voronoi tessellation were generated for each array and measures derived from these constructs used to analyze the arrays' spatial organization. As others have shown previously with a less complete statistical characterization, we found that the arrays of OFF- and ON-center β-cells have similar spatial properties and are more regular than the array of all β-cells. We developed a model to simulate cellular arrays with spatial properties like those of arrays of β-cells. A good fit between model and real arrays was found when the model assumed an explicit spatial dependence between the placement of OFF- and ON-center cells. We propose therefore that a single array of β-cells formed of both OFF- and ON-center cells is consistent with the data currently available for β-cell somatic arrays.

We give formulae for different types of contact distribution functions for stationary (not necessarily Poisson) Voronoi tessellations in ℝd in terms of the Palm void probabilities of the generating point process. Moreover, using the well-known relationship between the linear contact distribution and the chord length distribution we derive a closed form expression for the mean chord length in terms of the two-point Palm distribution and the pair correlation function of the generating point process. The results obtained are specified for Voronoi tessellations generated by Poisson cluster and Gibbsian processes, respectively.

We consider two independent homogeneous Poisson processes Π0 and Π1 in the plane with intensities λ0 and λ1, respectively. We study additive functionals of the set of Π0-particles within a typical Voronoi Π1-cell. We find the first and the second moments of these variables as well as upper and lower bounds on their distribution functions, implying an exponential asymptotic behavior of their tails. Explicit formulae are given for the number and the sum of distances from Π0-particles to the nucleus within a typical Voronoi Π1-cell.

Families of Poisson processes defined on general state spaces and with the intensity measure scaled by a positive parameter are investigated. In particular, mean value relations with respect to the scale parameter are established and used to derive various Gamma-type results for certain geometric characteristics determined by finite subprocesses. In particular, we deduce Miles' complementary theorem. Applications of the results within stochastic geometry and particularly for random tessellations are discussed.