Second when considering the Poisson point process
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 large-deviation analysis, we assume that the random environment is not only stabilizing, but in fact -dependent.
Model definition and main results Loosely speaking, Cox point processes are Poisson point processes in a random environment. More precisely, the random environment is given by a random interleukins in the space of Borel measures on equipped with the usual evaluation -algebra. Throughout the manuscript we assume that is stationary, but at this point we do not impose any additional conditions. In particular, could be an absolutely continuous or singular random intensity measure. Nevertheless, in some of the presented results, completely different behavior will appear.
Then, let be a Cox process in with stationary intensity measure where and . That is, conditioned on , the point process is a Poisson point process with intensity measure , see Fig. 1. To study continuum percolation on , we work with the Gilbert graph on the vertex set where two points are connected by an edge if their distance is less than a connection threshold . The graph percolates if it contains an infinite connected component.
Proof of phase transitions The main idea is to introduce a renormalization scheme reducing the continuum percolation problem to a dependent lattice percolation problem. To make this work, we rely crucially on the stabilization assumption. It allows us to make use of the standard -dependent percolation arguments presented in [19, Theorem 0.0].
To begin with, we prove the meta results announced in Section 2.3.3.
Acknowledgments We thank D. Ahlberg and J. Tykesson for the recent and highly valuable Ref. . Moreover, we thank an anonymous referee whose suggestions and comments improved the quality of the manuscript substantially. In particular, one of the remarks lead to a more general notion of asymptotic essential connectedness that is now also applicable to the super-critical Poisson–Boolean model. This research was supported by the Leibniz program Probabilistic Methods for Mobile Ad-Hoc Networks, by The Danish Council for Independent Research, Natural Sciences, grant DFF-7014-00074 Statistics for Point Processes in Space and Beyond, by the Centre for Stochastic Geometry and Advanced Bioimaging, funded by grant 8721 from the Villum Foundation, Denmark and by Orange S.A., France grant CRE G09292. We thank Nila Novita Gafur for her contribution in the simulation section.
Introduction Feature screening can effectively reduce ultrahigh dimensionality and therefore has attracted considerable attention in the recent literature. Fan and Lv  proposed a marginal screening procedure for ultrahigh-dimensional Gaussian linear model, and further showed that marginal screening procedures may possess a sure screening property under certain conditions. Feature screening procedures for varying-coefficient models (VCM) with ultrahigh-dimensional covariates have been proposed in the literature. Liu et al.  developed a sure independence screening (SIS) procedure for ultrahigh-dimensional VCM by taking conditional Pearson correlation coefficients as a marginal utility for ranking the importance of predictors. Fan et al.  proposed an SIS procedure for ultrahigh-dimensional VCM by extending B-spline techniques in Fan et al.  for additive models. Xia et al.  further extended the SIS procedure proposed in  to generalized varying-coefficient models (GVCM). Cheng et al.  proposed a forward variable selection procedure for ultrahigh-dimensional VCM based on techniques related to B-splines regression and grouped variable selection. Song et al.  extended the procedure in  to longitudinal data without taking into account within-subject correlation, while Chu et al.  proposed an SIS procedure for longitudinal data based on a weighted residual sum of squares to use within-subjection correlation to improve accuracy of feature screening. Kong et al.  proposed a new screening method that leaves a variable in the active set if it has, jointly with some other variables, a high canonical correlation with the response.