Motivated by an application in wireless telecommunication networks, we consider a two-type continuum-percolation problem involving a homogeneous Poisson point process of users and a stationary and ergodic point process of base stations. Starting from a randomly chosen point of the Poisson point process, we investigate the distribution of the minimum number of hops that are needed to reach some point of the base station process. In the supercritical regime of continuum percolation, we use the close relationship between Euclidean and chemical distance to identify the distributional limit of the rescaled minimum number of hops that are needed to connect a typical Poisson point to a point of the base station process as its intensity tends to 0. In particular, we obtain an explicit expression for the asymptotic probability that a typical Poisson point connects to a point of the base station process in a given number of hops.

A Poisson line tessellation is observed in the window Wρ := B(0, π-1/2ρ1/2) for ρ > 0. With each cell of the tessellation, we associate the inradius, which is the radius of the largest ball contained in the cell. Using the Poisson approximation, we compute the limit distributions of the largest and smallest order statistics for the inradii of all cells whose nuclei are contained in Wρ as ρ goes to ∞. We additionally prove that the limit shape of the cells minimising the inradius is a triangle.

In the full rectangular version of Gilbert's planar tessellation (see Gilbert (1967), Mackisack and Miles (1996), and Burridge et al. (2013)), lines extend either horizontally (with east- and west-growing rays) or vertically (north- and south-growing rays) from seed points which form a stationary Poisson point process, each ray stopping when it meets another ray that has blocked its path. In the half-Gilbert rectangular version, east- and south-growing rays, whilst having the potential to block each other, do not interact with west and north rays, and vice versa. East- and south-growing rays have a reciprocality of blocking, as do west and north. In this paper we significantly expand and simplify the half-Gilbert analytic results that we gave in Burridge et al. (2013). We also show how the idea of reciprocality of blocking between rays can be used in a much wider context, with rays not necessarily orthogonal and with seeds that produce more than two rays.

We consider an extension of the Poisson hail model where the service speed is either 0 or ∞ at each point of the Euclidean space. We use and develop tools pertaining to sub-additive ergodic theory in order to establish shape theorems for the growth of the ice-heap under light tail assumptions on the hailstone characteristics. The asymptotic shape depends on the statistics of the hailstones, the intensity of the underlying Poisson point process, and on the geometrical properties of the zero speed set.

Working in the infinite plane R2, consider a Poisson process of black points with intensity 1, and an independent Poisson process of red points with intensity λ. We grow a disc around each black point until it hits the nearest red point, resulting in a random configuration Aλ, which is the union of discs centered at the black points. Next, consider a fixed disc of area n in the plane. What is the probability pλ(n) that this disc is covered by Aλ? We prove that if λ3nlogn = y then, for sufficiently large n, e-8π2y ≤ pλ(n) ≤ e-2π2y/3. The proofs reveal a new and surprising phenomenon, namely, that the obstructions to coverage occur on a wide range of scales.

In a $d$ -dimensional convex body $K$ random points $X_{0},\ldots ,X_{d}$ are chosen. Their convex hull is a random simplex. The expected volume of a random simplex is monotone under set inclusion if $K\subset L$ implies that the expected volume of a random simplex in $K$ is smaller than the expected volume of a random simplex in $L$ . Continuing work of Rademacher [On the monotonicity of the expected volume of a random simplex. Mathematika58 (2012), 77–91], it is shown that moments of the volume of random simplices are, in general, not monotone under set inclusion.

This paper concerns the asymptotic behavior of a random variable W λ resulting from the summation of the functionals of a Gibbsian spatial point process over windows Q λ ↑ ℝ d . We establish conditions ensuring that W λ has volume order fluctuations, i.e. they coincide with the fluctuations of functionals of Poisson spatial point processes. We combine this result with Stein's method to deduce rates of a normal approximation for W λ as λ → ∞. Our general results establish variance asymptotics and central limit theorems for statistics of random geometric and related Euclidean graphs on Gibbsian input. We also establish a similar limit theory for claim sizes of insurance models with Gibbsian input, the number of maximal points of a Gibbsian sample, and the size of spatial birth-growth models with Gibbsian input.

We derive Laplace transform identities for the volume content of random stopping sets based on Poisson point processes. Our results are based on anticipating Girsanov identities for Poisson point processes under a cyclic vanishing condition for a finite difference gradient. This approach does not require classical assumptions based on set-indexed martingales and the (partial) ordering of index sets. The examples treated focus on stopping sets in finite volume, and include the random missed volume of Poisson convex hulls.

For a class of cell division processes in the Euclidean space ℝd , spatial consistency is investigated. This addresses the problem whether the distribution of the generated structures, restricted to a bounded set V, depends on the choice of a larger region W ⊃ V where the construction of the cell division process is performed. This can also be understood as the problem of boundary effects in the cell division procedure. It is known that the STIT tessellations are spatially consistent. In the present paper it is shown that, within a reasonable wide class of cell division processes, the STIT tessellations are the only ones that are consistent.

We provide a characterization of realisable set covariograms, bringing a rigorous yet abstract solution to the S 2 problem in materials science. Our method is based on the covariogram functional for random measurable sets (RAMS) and on a result about the representation of positive operators on a noncompact space. RAMS are an alternative to the classical random closed sets in stochastic geometry and geostatistics, and they provide a weaker framework that allows the manipulation of more irregular functionals, such as the perimeter. We therefore use the illustration provided by the S 2 problem to advocate the use of RAMS for solving theoretical problems of a geometric nature. Along the way, we extend the theory of random measurable sets, and in particular the local approximation of the perimeter by local covariograms.

Let $K$ be a convex body in $\mathbb{R}^{d}$ which slides freely in a ball. Let $K^{(n)}$ denote the intersection of $n$ closed half-spaces containing $K$ whose bounding hyperplanes are independent and identically distributed according to a certain prescribed probability distribution. We prove an asymptotic formula for the expectation of the difference of the volumes of $K^{(n)}$ and $K$, and an asymptotic upper bound on the variance of the volume of $K^{(n)}$. We obtain these asymptotic formulas by proving results for weighted mean width approximations of convex bodies that admit a rolling ball by inscribed random polytopes and then using polar duality to convert them into statements about circumscribed random polytopes.

We consider Euclidean first passage percolation on a large family of connected random geometric graphs in the d-dimensional Euclidean space encompassing various well-known models from stochastic geometry. In particular, we establish a strong linear growth property for shortest-path lengths on random geometric graphs which are generated by point processes. We consider the event that the growth of shortest-path lengths between two (end) points of the path does not admit a linear upper bound. Our linear growth property implies that the probability of this event tends to zero sub-exponentially fast if the direct (Euclidean) distance between the endpoints tends to infinity. Besides, for a wide class of stationary and isotropic random geometric graphs, our linear growth property implies a shape theorem for the Euclidean first passage model defined by such random geometric graphs. Finally, this shape theorem can be used to investigate a problem which is considered in structural analysis of fixed-access telecommunication networks, where we determine the limiting distribution of the length of the longest branch in the shortest-path tree extracted from a typical segment system if the intensity of network stations converges to 0.

Gaussian particles provide a flexible framework for modelling and simulating three-dimensional star-shaped random sets. In our framework, the radial function of the particle arises from a kernel smoothing, and is associated with an isotropic random field on the sphere. If the kernel is a von Mises-Fisher density, or uniform on a spherical cap, the correlation function of the associated random field admits a closed form expression. The Hausdorff dimension of the surface of the Gaussian particle reflects the decay of the correlation function at the origin, as quantified by the fractal index. Under power kernels we obtain particles with boundaries of any Hausdorff dimension between 2 and 3.

Vertices arrive sequentially in space and are joined to existing vertices at random according to a preferential rule combining degree and spatial proximity. We investigate phase transitions in the resulting graph as the relative strengths of these two components of the attachment rule are varied.

Previous work of one of the authors showed that when the geometric component is weak, the limiting degree sequence mimics the standard Barabási-Albert preferential attachment model. We show that at the other extreme, in the case of a sufficiently strong geometric component, the limiting degree sequence mimics a purely geometric model, the on-line nearest-neighbour graph, for which we prove some extensions of known results. We also show the presence of an intermediate regime, with behaviour distinct from both the on-line nearest-neighbour graph and the Barabási-Albert model; in this regime, we obtain a stretched exponential upper bound on the degree sequence.

Consider a real-valued discrete-time stationary and ergodic stochastic process, called the noise process. For each dimension n, we can choose a stationary point process in ℝn and a translation invariant tessellation of ℝn . Each point is randomly displaced, with a displacement vector being a section of length n of the noise process, independent from point to point. The aim is to find a point process and a tessellation that minimizes the probability of decoding error, defined as the probability that the displaced version of the typical point does not belong to the cell of this point. We consider the Shannon regime, in which the dimension n tends to ∞, while the logarithm of the intensity of the point processes, normalized by dimension, tends to a constant. We first show that this problem exhibits a sharp threshold: if the sum of the asymptotic normalized logarithmic intensity and of the differential entropy rate of the noise process is positive, then the probability of error tends to 1 with n for all point processes and all tessellations. If it is negative then there exist point processes and tessellations for which this probability tends to 0. The error exponent function, which denotes how quickly the probability of error goes to 0 in n, is then derived using large deviations theory. If the entropy spectrum of the noise satisfies a large deviations principle, then, below the threshold, the error probability goes exponentially fast to 0 with an exponent that is given in closed form in terms of the rate function of the noise entropy spectrum. This is obtained for two classes of point processes: the Poisson process and a Matérn hard-core point process. New lower bounds on error exponents are derived from this for Shannon's additive noise channel in the high signal-to-noise ratio limit that hold for all stationary and ergodic noises with the above properties and that match the best known bounds in the white Gaussian noise case.

Let X 1, X 2, …, X n be independent and uniformly distributed random variables in the unit square [0, 1]2, and let L(X 1, …, X n ) be the length of the shortest traveling salesman path through these points. In 1959, Beardwood, Halton and Hammersley proved the existence of a universal constant β such that limn→∞ n−1/2 L(X 1, …, X n ) = β almost surely. The best bounds for β are still those originally established by Beardwood, Halton and Hammersley, namely 0.625 ≤ β ≤ 0.922. We slightly improve both upper and lower bounds.

We study the asymptotics of a Markovian system of N ≥ 3 particles in [0, 1]d in which, at each step in discrete time, the particle farthest from the current centre of mass is removed and replaced by an independent U[0, 1]d random particle. We show that the limiting configuration contains N − 1 coincident particles at a random location ξN ∈ [0, 1]d . A key tool in the analysis is a Lyapunov function based on the squared radius of gyration (sum of squared distances) of the points. For d = 1, we give additional results on the distribution of the limit ξN , showing, among other things, that it gives positive probability to any nonempty interval subset of [0, 1], and giving a reasonably explicit description in the smallest nontrivial case, N = 3.

This paper provides tools for the study of the Dirichlet random walk in R d . We compute explicitly, for a number of cases, the distribution of the random variable W using a form of Stieltjes transform of W instead of the Laplace transform, replacing the Bessel functions with hypergeometric functions. This enables us to simplify some existing results, in particular, some of the proofs by Le Caër (2010), (2011). We extend our results to the study of the limits of the Dirichlet random walk when the number of added terms goes to ∞, interpreting the results in terms of an integral by a Dirichlet process. We introduce the ideas of Dirichlet semigroups and Dirichlet infinite divisibility and characterize these infinite divisible distributions in the sense of Dirichlet when they are concentrated on the unit sphere of R d .

We consider a stationary Poisson hyperplane process with given directional distribution and intensity in d-dimensional Euclidean space. Generalizing the zero cell of such a process, we fix a convex body K and consider the intersection of all closed halfspaces bounded by hyperplanes of the process and containing K. We study how well these random polytopes approximate K (measured by the Hausdorff distance) if the intensity increases, and how this approximation depends on the directional distribution in relation to properties of K.

In this paper we consider the stationary Poisson Boolean model with spherical grains and propose a family of nonparametric estimators for the radius distribution. These estimators are based on observed distances and radii, weighted in an appropriate way. They are ratio unbiased and asymptotically consistent for a growing observation window. We show that the asymptotic variance exists and is given by a fairly explicit integral expression. Asymptotic normality is established under a suitable integrability assumption on the weight function. We also provide a short discussion of related estimators as well as a simulation study.