We derive an asymptotic expansion for the critical percolation density of the random connection model as the dimension of the encapsulating space tends to infinity. We calculate rigorously the first expansion terms for the Gilbert disk model, the hyper-cubic model, the Gaussian connection kernel, and a coordinate-wise Cauchy kernel.

We investigate expansions for connectedness functions in the random connection model of continuum percolation in powers of the intensity. Precisely, we study the pair-connectedness and the direct-connectedness functions, related to each other via the Ornstein–Zernike equation. We exhibit the fact that the coefficients of the expansions consist of sums over connected and 2-connected graphs. In the physics literature, this is known to be the case more generally for percolation models based on Gibbs point processes and stands in analogy to the formalism developed for correlation functions in liquid-state statistical mechanics.

We find a representation of the direct-connectedness function and bounds on the intensity which allow us to pass to the thermodynamic limit. In some cases (e.g., in high dimensions), the results are valid in almost the entire subcritical regime. Moreover, we relate these expansions to the physics literature and we show how they coincide with the expression provided by the lace expansion.

Consider a graph on randomly scattered points in an arbitrary space, with any two points x, y connected with probability ϕ(x, y). Suppose the number of points is large but the mean number of isolated points is O(1). We give general criteria for the latter to be approximately Poisson distributed. More generally, we consider the number of vertices of fixed degree, the number of components of fixed order, and the number of edges. We use a general result on Poisson approximation by Stein's method for a set of points selected from a Poisson point process. This method also gives a good Poisson approximation for U-statistics of a Poisson process.

In the first part of this paper we consider a general stationary subcritical cluster model in ℝd. The associated pair-connectedness function can be defined in terms of two-point Palm probabilities of the underlying point process. Using Palm calculus and Fourier theory we solve the Ornstein–Zernike equation (OZE) under quite general distributional assumptions. In the second part of the paper we discuss the analytic and combinatorial properties of the OZE solution in the special case of a Poisson-driven random connection model.

When modeling the spread of infectious diseases, it is important to incorporate riskbehavior of individuals in a considered population. Not only risk behavior, but also thenetwork structure created by the relationships among these individuals as well as thedynamical rules that convey the spread of the disease are the key elements in predictingand better understanding the spread. In this work we propose the weighted randomconnection model, where each individual of the population is characterized by twoparameters: its position and risk behavior. A goal is to model the effect that theprobability of transmissions among individuals increases in the individual risk factors,and decays in their Euclidean distance. Moreover, the model incorporates a combined riskbehavior function for every pair of the individuals, through which the spread can bedirectly modeled or controlled. We derive conditions for the existence of an outbreak ofinfectious diseases in this model. Our main result is the almost sure existence of aninfinite component in the weighted random connection model. We use results on the randomconnection model and site percolation in Z2.