Continuum percolation for Cox point processes

Abstract

We investigate continuum percolation for Cox point processes, that is, Poisson point processes driven by random intensity measures. First, we derive sufficient conditions for the existence of non-trivial sub-and super-critical percolation regimes based on the notion of stabilization. Second, we give asymptotic expressions for the percolation probability in largeradius, high-density and coupled regimes. In some regimes, we find universality, whereas in others, a sensitive dependence on the underlying random intensity measure survives.celebrated concept of stabilization [17,[26][27][28]31] suffices to guarantee the existence of a subcritical phase. In contrast, for the existence of a super-critical phase, stabilization alone is not enough since percolation is impossible unless the support of the random measure has sufficiently good connectivity properties itself. Hence, our proof for the existence of a super-critical phase relies on a variant of the notion of asymptotic essential connectedness from [2].Second, when considering the Poisson point process, the high-density or large-radius limit of the percolation probability tends to 1 exponentially fast and is governed by the isolation probability. In the random environment, the picture is more subtle since the regime of a large radius is no longer equivalent to that of a high density. Since we rely on a refined largedeviation analysis, we assume that the random environment is not only stabilizing, but in fact b-dependent.Since the high-density and the large-radius limit are no longer equivalent, this opens up the door to an analysis of coupled limits. As we shall see, the regime of a large radius and low density is of highly averaging nature and therefore results in a universal limiting behavior. On the other hand, in the converse limit the geometric structure of the random environment remains visible in the limit. In particular, a different scaling balance between the radius and density is needed when dealing with absolutely continuous and singular random measures, respectively. Finally, we illustrate our results with specific examples and simulations. Model definition and main resultsLoosely speaking, Cox point processes are Poisson point processes in a random environment. More precisely, the random environment is given by a random element Λ in the space M of Borel measures on R d equipped with the usual evaluation σ-algebra. Throughout the manuscript we assume that Λ is stationary, but at this point we do not impose any additional conditions. In particular, Λ could be an absolutely continuous or singular random intensity measure. Nevertheless, in some of the presented results, completely different behavior will appear. Example 2.1 (Absolutely continuous environment). Let Λ(dx) = x dx with = { x } x∈R d a stationary non-negative random field. For example, this includes random measures modulated by a random closed set Ξ, [7, Section 5.2.2]. Here, x = λ 1 1{x ∈ Ξ} + λ 2 1{x ∈ Ξ} with λ 1 , λ 2 ≥ 0. Another example are random measures induced by shot-noise fields, [7...

Signal-to-interference ratio percolation for Cox point processes

“…A Cox point process X λ with intensity λΛ is characterized by the property that conditional on Λ, X λ is a Poisson point process with intensity λΛ. In [HJC18], it was shown that under certain stabilization and connectedness conditions on Λ, 0 < λ c (r) < ∞ holds. More precisely, λ c (r) > 0 if Λ is stabilizing and λ c (r) < ∞ under the stronger assumption that Λ is asymptotically essentially connected.…”

“…According to [HJC18], the most important examples of Λ for telecommunication are given by a stationary tessellation process, e.g., a Poisson-Voronoi, Poisson-Delaunay or Poisson line tessellation. The edge set of such a tessellation process can be used for modelling a telecommunication network on a street system, where the points of the Cox point process are the users, situated on the streets.…”

Exponential Moments for Planar Tessellations

“…exists, see [HJC18,Lemma 6.1]. This can be used for example to establish the limiting behavior of the percolation probability for the Boolean model with large radii based on Cox point processes where the intensity measure is given by |S ∩ dx|, see [HJC18].…”

The Influence of Canyon Shadowing on Device-to-Device Connectivity in Urban Scenario

“…There is a possible theoretical gap between H * ≥ H c and the critical value H c ≈ 0.743 found in Section IV-B, however our simulations suggest that H * ≈ H c . A rigorous proof of the above result follows the approach developed in [15]. As the goal of this paper is more about giving numerical estimates, and due to space constraints, we only give a rough sketch of the proof.…”

“…The main problem faced in the study of percolation in a random environment (PVT street system in our case) is the spatial dependence of the environment. By the stabilization property [15] of the PVT, the configuration of the network environment in a given observation window only depends on a bounded region including the observation window with high probability. In other words, if two observed regions of the network are distant enough, they are independent.…”