We develop a computational approximation to the intensity of a Gibbs spatial point process having interactions of any order. Limit theorems from stochastic geometry, and small-sample probabilities estimated once and for all by an extensive simulation study, are combined with scaling properties to form an approximation to the moment generating function of the sufficient statistic under a Poisson process. The approximate intensity is obtained as the solution of a self-consistency equation.

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.

Double-stranded DNA molecules can be damaged by enzymic action or radiation, in a manner which creates randomly-located single-stranded breaks (nicks). The accumulation of these leads eventually to the double-stranded breakage of the molecule, because two opposite-strand nicks within a critical distance of each other establish conditions for breakage. We study the random variable N, defined as the number of nicks needed for double-stranded breakage to occur. We develop an asymptotic theory which is needed for practical computations.