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...

We prove lower large deviations for geometric functionals in sparse, critical and dense regimes. Our results are tailored for functionals with nonexisting exponential moments, for which standard large deviation theory is not applicable. The primary tool of the proofs is a sprinkling technique that, adapted to the considered functionals, ensures a certain boundedness. This substantially generalizes previous approaches to tackle lower tails with sprinkling. Applications include subgraph counts, persistent Betti numbers and edge lengths based on a sparse random geometric graph, power-weighted edge lengths of a k-nearest neighbor graph as well as power-weighted spherical contact distances in a critical regime and volumes of k-nearest neighbor balls in a dense regime.

Percolation properties of the dead leaves model, also known as confetti percolation, are considered. More precisely, we prove that the critical probability for confetti percolation with square-shaped leaves is 1/2. This result is related to a question of Benjamini and Schramm concerning disk-shaped leaves and can be seen as a variant of the Harris-Kesten theorem for bond percolation. The proof is based on techniques developed by Bollobás and Riordan to determine the critical probability for Voronoi and Johnson-Mehl percolation.

We introduce tests for the goodness of fit of point patterns via methods from topological data analysis. More precisely, the persistent Betti numbers give rise to a bivariate functional summary statistic for observed point patterns that is asymptotically Gaussian in large observation windows. We analyze the power of tests derived from this statistic on simulated point patterns and compare its performance with global envelope tests. Finally, we apply the tests to a point pattern from an application context in neuroscience. As the main methodological contribution, we derive sufficient conditions for a functional central limit theorem on bounded persistent Betti numbers of point processes with exponential decay of correlations.

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