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...

Dedicated to A. van Enter on the occasion of his 60th birthdayWe consider a class of discrete q-state spin models defined in terms of a translation-invariant quasilocal specification with discrete clock-rotation invariance which have extremal Gibbs measures µ ′ ϕ labeled by the uncountably many values of ϕ in the one-dimensional sphere (introduced by van Enter, Opoku, Külske [J. Phys. A 44 (2011) 475002, 11]). In the present paper we construct an associated Markov jump process with quasilocal rates whose semigroup (St) t≥0 acts by a continuous rotation St(µ ′ ϕ ) = µ ′ ϕ+t . As a consequence our construction provides examples of interacting particle systems with unique translation-invariant invariant measure, which is not long-time limit of all starting measures, answering an old question (compare Liggett [Interacting Particle Systems (1985) Springer], question four, Chapter one). The construction of this particle system is inspired by recent conjectures of Maes and Shlosman about the intermediate temperature regime of the nearest-neighbor clock model. We define our generator of the interacting particle system as a (noncommuting) sum of the rotation part and a Glauber part.Technically the paper rests on the control of the spread of weak nonlocalities and relative entropy-methods, both in equilibrium and dynamically, based on Dobrushin-uniqueness bounds for conditional measures.

Abstract. We consider the continuum Widom-Rowlinson model under independent spin-flip dynamics and investigate whether and when the time-evolved point process has an (almost) quasilocal specification (Gibbs-property of the time-evolved measure). Our study provides a first analysis of a Gibbs-non-Gibbs transition for point particles in Euclidean space. We find a picture of loss and recovery, in which even more regularity is lost faster than it is for time-evolved spin models on lattices.We show immediate loss of quasilocality in the percolation regime, with full measure of discontinuity points for any specification. For the color-asymmetric percolating model, there is a transition from this non-a.s. quasilocal regime back to an everywhere Gibbsian regime. At the sharp reentrance time tG > 0 the model is a.s. quasilocal. For the color-symmetric model there is no reentrance. On the constructive side, for all t > tG, we provide everywhere quasilocal specifications for the time-evolved measures and give precise exponential estimates on the influence of boundary condition.

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.

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