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.

Denote the Palm measure of a homogeneous Poisson process H λ with two points 0 and x by P0,x . We prove that there exists a constant μ ≥ 1 such that P0,x (D(0, x) / μ||x||2 ∉ (1 − ε, 1 + ε) | 0, x ∈ C ∞) exponentially decreases when ||x||2 tends to ∞, where D(0, x) is the graph distance between 0 and x in the infinite component C ∞ of the random geometric graph G(H λ; 1). We derive a large deviation inequality for an asymptotic shape result. Our results have applications in many fields and especially in wireless sensor networks.

We consider a family of long-range percolation models (G p )p>0 on ℤd that allow dependence between edges and have the following connectivity properties for p ∈ (1/d, ∞): (i) the degree distribution of vertices in G p has a power-law distribution; (ii) the graph distance between points x and y is bounded by a multiple of logpd logpd |x - y | with probability 1 - o(1); and (iii) an adversary can delete a relatively small number of nodes from G p (ℤd ∩ [0, n]d ), resulting in two large, disconnected subgraphs.

The aim of this paper is to extend the well-known asymptotic shape result for first-passage percolation on $\mathbb{Z}^d$ to first-passage percolation on a random environment given by the infinite cluster of a supercritical Bernoulli percolation model. We prove the convergence of the renormalized set of wet vertices to a deterministic shape that does not depend on the realization of the infinite cluster.As a special case of our result, we obtain an asymptotic shape theorem for the chemical distance in supercritical Bernoulli percolation.We also prove a flat edge result in the case of dimension 2. Various examples are also given.