Poisson Boolean Model Research Articles

Phase transitions and noise sensitivity on the Poisson space via stopping sets and decision trees

AbstractProofs of sharp phase transition and noise sensitivity in percolation have been significantly simplified by the use of randomized algorithms, via the OSSS inequality (proved by O'Donnell et al. and the Schramm–Steif inequality for the Fourier‐Walsh coefficients of functions defined on the Boolean hypercube. In this article, we prove intrinsic versions of the OSSS and Schramm–Steif inequalities for functionals of a general Poisson process, and apply these new estimates to deduce sufficient conditions—expressed in terms of randomized stopping sets—yielding sharp phase transitions, quantitative noise sensitivity, exceptional times and bounds on critical windows for monotonic Boolean Poisson functions. Our analysis is based on a new general definition of “stopping set”, not requiring any topological property for the underlying measurable space, as well as on the new concept of a “continuous‐time decision tree”, for which we establish several fundamental properties. We apply our findings to the ‐percolation of the Poisson Boolean model and to the Poisson‐based confetti percolation with bounded random grains. In these two models, we reduce the proof of sharp phase transitions for percolation, and of noise sensitivity for crossing events, to the construction of suitable randomized stopping sets and the computation of one‐arm probabilities at criticality. This enables us to settle an open problem suggested by Ahlberg et al. (a special case of which was conjectured earlier by Ahlberg et al. on noise sensitivity of crossing events for the planar Poisson Boolean model with random balls whose radius distribution has finite ‐moments and also show the same for planar confetti percolation model with bounded random balls. We also prove that critical probability is for the planar confetti percolation model with bounded, ‐rotation invariant and reflection invariant random grains. Such a result was conjectured by Benjamini and Schramm in the case of fixed balls and proved by Müller, Hirsch and Ghosh and Roy in the case of balls, boxes and random boxes, respectively; our results contain all previous findings as special cases.

Percolation of worms

We introduce a new correlated percolation model on the d-dimensional lattice Zd called the random length worms model. Assume given a probability distribution on the set of positive integers (the length distribution) and v∈(0,∞) (the intensity parameter). From each site of Zd we start POI(v) independent simple random walks with this length distribution. We investigate the connectivity properties of the set Sv of sites visited by this cloud of random walks. It is easy to show that if the second moment of the length distribution is finite then Sv undergoes a percolation phase transition as v varies. Our main contribution is a sufficient condition on the length distribution which guarantees that Sv percolates for all v>0 if d≥5. E.g., if the probability mass function of the length distribution is m(ℓ)=c⋅ln(ln(ℓ))ɛ/(ℓ3ln(ℓ))1[ℓ≥ℓ0]for some ℓ0>ee and ɛ>0 then Sv percolates for all v>0. Note that the second moment of this length distribution is only “barely” infinite. In order to put our result in the context of earlier results about similar models (e.g., finitary random interlacements, loop percolation, Bernoulli hyper-edge percolation, Poisson Boolean model, ellipses percolation, etc.), we define a natural family of percolation models called the Poisson zoo and argue that the percolative behaviour of the random length worms model is quite close to being “extremal” in this family of models.

Left–right crossings in the Miller–Abrahams random resistor network and in generalized Boolean models

We consider random graphs G built on a homogeneous Poisson point process on Rd, d≥2, with points x marked by i.i.d. random variables Ex. Fixed a symmetric function h(⋅,⋅), the vertexes of G are given by points of the Poisson point process, while the edges are given by pairs {x,y} with x≠y and |x−y|≤h(Ex,Ey). We call GPoissonh-generalized Boolean model, as one recovers the standard Poisson Boolean model by taking h(a,b)≔a+b and Ex≥0. Under general conditions, we show that in the supercritical phase the maximal number of vertex-disjoint left–right crossings in a box of size n is lower bounded by Cnd−1 apart from an event of exponentially small probability. As special applications, when the marks are non-negative, we consider the Poisson Boolean model and its generalization to h(a,b)=(a+b)γ with γ>0, the weight-dependent random connection models with max-kernel and with min-kernel and the graph obtained from the Miller–Abrahams random resistor network in which only filaments with conductivity lower bounded by a fixed positive constant are kept.

Homological percolation and the Euler characteristic.

In this paper we study the connection between the zeros of the expected Euler characteristic curve and the phenomenon which we refer to as homological percolation-the formation of "giant" cycles in persistent homology, which is intimately related to classical notions of percolation. We perform an experimental study that covers four different models: site percolation on the cubical and permutahedral lattices, the Poisson-Boolean model, and Gaussian random fields. All the models are generated on the flat torus T^{d} for d=2,3,4. The simulation results strongly indicate that the zeros of the expected Euler characteristic curve approximate the critical values for homological percolation. Our results also provide some insight about the approximation error. Further study of this connection could have powerful implications both in the study of percolation theory and in the field of topological data analysis.

The existence phase transition for scale invariant Poisson random fractal models

In this paper we study the existence phase transition of scale invariant random fractal models. We determine the exact value of the critical point of this phase transition for all models satisfying some weak assumptions. In addition, we show that for a large subclass, the fractal model is in the empty phase at the critical point. This subclass of models includes the scale invariant Poisson Boolean model and the Brownian loop soup. In contrast to earlier results in the literature, we do not need to restrict our attention to random fractal models generated by open sets.

On the critical threshold for continuum AB percolation

Abstract Consider a bipartite random geometric graph on the union of two independent homogeneous Poisson point processes in d-space, with distance parameter r and intensities λ,μ. For any λ>0 we consider the percolation threshold μc(λ) associated to the parameter μ. Denoting by λc the percolation threshold for the standard Poisson Boolean model with radii r, we show the lower bound μc(λ)≥clog(c∕(λ−λc)) for any λ>λc with c>0 a fixed constant. In particular, there is no phase transition in μ at the critical value of λ, that is, μc(λc) =∞.

Non-triviality of the vacancy phase transition for the Boolean model

In the spherical Poisson Boolean model, one takes the union of random balls centred on the points of a Poisson process in Euclidean $d$-space with $d \geq 2$. We prove that whenever the radius distribution has a finite $d$-th moment, there exists a strictly positive value for the intensity such that the vacant region percolates.

Invariance Under Quasi-isometries of Subcritical and Supercritical Behavior in the Boolean Model of Percolation

In this work we study the Poisson Boolean model of percolation in locally compact Polish metric spaces and we prove the invariance of subcritical and supercritical phases under mm-quasi-isometries. In other words, we prove that if the Poisson Boolean model of percolation is subcritical or supercritical (or exhibits phase transition) in a metric space M which is mm-quasi-isometric to a metric space N, then these phases also exist for the Poisson Boolean model of percolation in N. Then we apply these results to understand the phenomenon of phase transition in a large family of metric spaces. Indeed, we study the Poisson Boolean model of percolation in the context of Riemannian manifolds, in a large family of nilpotent Lie groups and in Cayley graphs. Also, we prove the existence of a subcritical phase in Gromov spaces with bounded growth at some scale.

Statistics for Poisson Models of Overlapping Spheres

In this paper we consider the stationary Poisson Boolean model with spherical grains and propose a family of nonparametric estimators for the radius distribution. These estimators are based on observed distances and radii, weighted in an appropriate way. They are ratio unbiased and asymptotically consistent for a growing observation window. We show that the asymptotic variance exists and is given by a fairly explicit integral expression. Asymptotic normality is established under a suitable integrability assumption on the weight function. We also provide a short discussion of related estimators as well as a simulation study.

Statistics for Poisson Models of Overlapping Spheres

In this paper we consider the stationary Poisson Boolean model with spherical grains and propose a family of nonparametric estimators for the radius distribution. These estimators are based on observed distances and radii, weighted in an appropriate way. They are ratio unbiased and asymptotically consistent for a growing observation window. We show that the asymptotic variance exists and is given by a fairly explicit integral expression. Asymptotic normality is established under a suitable integrability assumption on the weight function. We also provide a short discussion of related estimators as well as a simulation study.

Noise sensitivity in continuum percolation

We prove that the Poisson Boolean model, also known as the Gilbert disc model, is noise sensitive at criticality. This is the first such result for a Continuum Percolation model, and the first which involves a percolation model with critical probability p c ≠ 1/2. Our proof uses a version of the Benjamini-Kalai-Schramm Theorem for biased product measures. A quantitative version of this result was recently proved by Keller and Kindler. We give a simple deduction of the non-quantitative result from the unbiased version. We also develop a quite general method of approximating Continuum Percolation models by discrete models with p c bounded away from zero; this method is based on an extremal result on non-uniform hypergraphs.

Percolation and Connectivity in AB Random Geometric Graphs

Given two independent Poisson point processes Φ(1), Φ(2) in , the AB Poisson Boolean model is the graph with the points of Φ(1) as vertices and with edges between any pair of points for which the intersection of balls of radius 2r centered at these points contains at least one point of Φ(2). This is a generalization of the AB percolation model on discrete lattices. We show the existence of percolation for all d ≥ 2 and derive bounds for a critical intensity. We also provide a characterization for this critical intensity when d = 2. To study the connectivity problem, we consider independent Poisson point processes of intensities n and τn in the unit cube. The AB random geometric graph is defined as above but with balls of radius r. We derive a weak law result for the largest nearest-neighbor distance and almost-sure asymptotic bounds for the connectivity threshold.

Percolation and Connectivity in AB Random Geometric Graphs

Given two independent Poisson point processes Φ(1), Φ(2) in , the AB Poisson Boolean model is the graph with the points of Φ(1) as vertices and with edges between any pair of points for which the intersection of balls of radius 2r centered at these points contains at least one point of Φ(2). This is a generalization of the AB percolation model on discrete lattices. We show the existence of percolation for all d ≥ 2 and derive bounds for a critical intensity. We also provide a characterization for this critical intensity when d = 2. To study the connectivity problem, we consider independent Poisson point processes of intensities n and τn in the unit cube. The AB random geometric graph is defined as above but with balls of radius r. We derive a weak law result for the largest nearest-neighbor distance and almost-sure asymptotic bounds for the connectivity threshold.

Geometry of the Poisson Boolean model on a region of logarithmic width in the plane

Consider the region L = {(x, y): 0 ≤ y ≤ Clog(1 + x), x > 0} for a constant C > 0. We study the percolation and coverage properties of this region. For the coverage properties, we place a Poisson point process of intensity λ on the entire half space R + x R and associated with each Poisson point we place a box of a random side length ρ. Depending on the tail behaviour of the random variable ρ we exhibit a phase transition in the intensity for the eventual coverage of the region L. For the percolation properties, we place a Poisson point process of intensity λ on the region R 2. At each point of the process we centre a box of a random side length ρ. In the case ρ ≤ R for some fixed R > 0 we study the critical intensity λc of the percolation on L.

Geometry of the Poisson Boolean model on a region of logarithmic width in the plane

Consider the region L = {(x, y): 0 ≤ y ≤ Clog(1 + x), x > 0} for a constant C > 0. We study the percolation and coverage properties of this region. For the coverage properties, we place a Poisson point process of intensity λ on the entire half space R+ x R and associated with each Poisson point we place a box of a random side length ρ. Depending on the tail behaviour of the random variable ρ we exhibit a phase transition in the intensity for the eventual coverage of the region L. For the percolation properties, we place a Poisson point process of intensity λ on the region R2. At each point of the process we centre a box of a random side length ρ. In the case ρ ≤ R for some fixed R > 0 we study the critical intensity λc of the percolation on L.

On the visibility in well-behaved random sets in Euclidean space

In Calka et al. (2009), the decay of the probability of reaching distance at least r in some direction from a given point without hitting any ball, in the Poisson Boolean model of continuum percolation, was studied. The methods used in Calka et al. (2009) include coverage techniques, and the most precise results were obtained in dimension 2 . In this note, we strengthen some of the results obtained in Calka et al. (2009) to dimension 3 and higher and at the same time extend them to more general random sets.

Some properties for the largest component of random geometric graphs with applications in sensor networks

In this paper we consider the standard Poisson Boolean model of random geometric graphs G(\( \mathcal{H}_{\lambda ,s} \); 1) in ℝd and study the properties of the order of the largest component L1(G(\( \mathcal{H}_{\lambda ,s} \); 1)). We prove that E[L1(G(\( \mathcal{H}_{\lambda ,s} \); 1))] is smooth with respect to λ, and is derivable with respect to s. Also, we give the expression of these derivatives. These studies provide some new methods for the theory of the largest component of finite random geometric graphs (not asymptotic graphs as s → ∞) in the high dimensional space (d ≥ 2). Moreover, we investigate the convergence rate of E[L1(G(\( \mathcal{H}_{\lambda ,s} \); 1))]. These results have significance for theory development of random geometric graphs and its practical application. Using our theories, we construct and solve a new optimal energy-efficient topology control model of wireless sensor networks, which has the significance of theoretical foundation and guidance for the design of network layout.

On the coverage process of a moving point target in a non-uniform dynamic sensor field

We analyze the statistical properties of the k-coverage of a point-target moving in a straight line in a nonuniform dynamic sensor field. Sensor locations form a spatial point process. The environmental variation is captured by making the sensor locations form a non homogeneous spatial Poisson process with a fixed, spatially varying density function. The sensing areas of the sensors are circles of i.i.d. radii. The availability of each node is modeled by an independent, {0, 1}- valued, continuous time Markov chain. This gives a Markov-non homogeneous Poisson-Boolean model for which we perform a coverage analysis. We first obtain k-coverage of the target at an arbitrary time instant. We then obtain k-coverage statistics of the target during the time interval [0, T]. We also provide an asymptotically tight, closed form approximation for the duration for which the target is not k-covered in [0, T]. Numerical results illustrate the analysis. The environmental variation can also be captured by modeling the density function as a spatial random process resulting in the point process being a two-dimensional Cox process. For this model, we discuss issues in the coverage analysis.

A Bayesian Hierarchical Nonoverlapping Random Disc Growth Model

A methodology is proposed to efficiently model a random set via a multistage hierarchical Bayesian model. We define a NonOverlapping Random Disk Model (NORDM), which is similar in spirit to the well-known Poisson–Boolean model. This model is formulated in a conditional setting that facilitates Bayesian sampling of important parameters in the model. This framework can accommodate any object, not just those with disk shapes, although the model can be easily extended to include any known compact convex set instead of the disc (e.g., polygons or ellipses). We further propose a growth model that is conceptually simple and allows straightforward estimation of parameters, without the need for tedious calculations of hitting or inclusion probabilities. The model is applied to severe storm cell development as obtained from weather radar.

Limit Laws for k-Coverage of Paths by a Markov–Poisson–Boolean Model

Let P: = {X i } i≥1 be a stationary Poisson point process in ℜ d , {C i } i≥1 be a sequence of i.i.d. random sets in ℜ d , and be i.i.d. {0,1}-valued continuous time stationary Markov chains. We define the Markov–Poisson–Boolean model . 𝒞 t represents the coverage process at time t. We first obtain limit laws for k-coverage of an area at an arbitrary instant. We then obtain the limit laws for the k-coverage seen by a particle as it moves along a one-dimensional path.