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

2011 ◽  
Vol 43 (3) ◽  
pp. 616-635
Author(s):  
Amites Dasgupta ◽  
Rahul Roy ◽  
Anish Sarkar
Keyword(s):  
Phase Transition ◽  
Half Space ◽  
Point Process ◽  
Poisson Point Process ◽  
Random Variable ◽  
Boolean Model ◽  
Side Length ◽  
Tail Behaviour ◽  
Critical Intensity ◽  
Poisson Boolean Model

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.

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

2011 ◽  
Vol 43 (03) ◽  
pp. 616-635
Author(s):  
Amites Dasgupta ◽  
Rahul Roy ◽  
Anish Sarkar
Keyword(s):  
Phase Transition ◽  
Half Space ◽  
Point Process ◽  
Poisson Point Process ◽  
Random Variable ◽  
Boolean Model ◽  
Side Length ◽  
Tail Behaviour ◽  
Critical Intensity ◽  
Poisson Boolean Model

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.

scholarly journals On Laslett's transform for the Boolean model

2001 ◽  
Vol 33 (1) ◽  
pp. 1-5 ◽  
Author(s):  
A. D. Barbour ◽  
V. Schmidt
Keyword(s):  
Poisson Process ◽  
Point Process ◽  
Simple Proof ◽  
Poisson Point Process ◽  
Boolean Model ◽  
Random Sets ◽  
Convex Compact ◽  
Homogeneous Poisson Process ◽  
Vertical Strip ◽  
Cumulative Process

Consider the Boolean model in ℝ2, where the germs form a homogeneous Poisson point process with intensity λ and the grains are convex compact random sets. It is known (see, e.g., Cressie (1993, Section 9.5.3)) that Laslett's rule transforms the exposed tangent points of the Boolean model into a homogeneous Poisson process with the same intensity. In the present paper, we give a simple proof of this result, which is based on a martingale argument. We also consider the cumulative process of uncovered area in a vertical strip and show that a (linear) Poisson process with intensity λ can be embedded in it.

Critical intensities of Boolean models with different underlying convex shapes

2002 ◽  
Vol 34 (01) ◽  
pp. 48-57
Author(s):  
Rahul Roy ◽  
Hideki Tanemura
Keyword(s):  
Unit Area ◽  
Higher Dimensions ◽  
Boolean Model ◽  
Two Dimensions ◽  
Three Dimensions ◽  
Boolean Models ◽  
Regular Polytope ◽  
Minimum Value ◽  
Critical Intensity ◽  
Poisson Boolean Model

We consider the Poisson Boolean model of percolation where the percolating shapes are convex regions. By an enhancement argument we strengthen a result of Jonasson (2000) to show that the critical intensity of percolation in two dimensions is minimized among the class of convex shapes of unit area when the percolating shapes are triangles, and, for any other shape, the critical intensity is strictly larger than this minimum value. We also obtain a partial generalization to higher dimensions. In particular, for three dimensions, the critical intensity of percolation is minimized among the class of regular polytopes of unit volume when the percolating shapes are tetrahedrons. Moreover, for any other regular polytope, the critical intensity is strictly larger than this minimum value.

scholarly journals Asymptotics of Visibility in the Hyperbolic Plane

2013 ◽  
Vol 45 (02) ◽  
pp. 332-350
Author(s):  
Johan Tykesson ◽  
Pierre Calka
Keyword(s):  
Fixed Point ◽  
Point Process ◽  
Hyperbolic Plane ◽  
Poisson Point Process ◽  
Asymptotic Results ◽  
Critical Intensity

At each point of a Poisson point process of intensityλin the hyperbolic plane, center a ball of bounded random radius. Consider the probabilityPrthat, from a fixed point, there is some direction in which one can reach distancerwithout hitting any ball. It is known (see Benjamini, Jonasson, Schramm and Tykesson (2009)) that ifλis strictly smaller than a critical intensityλgvthenPrdoes not go to 0 asr → ∞. The main result in this note shows that in the caseλ=λgv, the probability of reaching a distance larger thanrdecays essentially polynomially, while ifλ>λgv, the decay is exponential. We also extend these results to various related models and we finally obtain asymptotic results in several situations.

scholarly journals Compound Poisson point processes, concentration and oracle inequalities

2019 ◽  
Vol 2019 (1) ◽  
Author(s):  
Huiming Zhang ◽  
Xiaoxu Wu
Keyword(s):  
Point Process ◽  
Poisson Point Process ◽  
Point Processes ◽  
Negative Binomial ◽  
Random Variable ◽  
Acta Math ◽  
Concentration Inequalities ◽  
Oracle Inequalities ◽  
Compound Poisson ◽  
Binomial Random Variable

AbstractThis note aims at presenting several new theoretical results for the compound Poisson point process, which follows the work of Zhang et al. (Insur. Math. Econ. 59:325–336, 2014). The first part provides a new characterization for a discrete compound Poisson point process (proposed by Aczél (Acta Math. Hung. 3(3):219–224, 1952)), it extends the characterization of the Poisson point process given by Copeland and Regan (Ann. Math. 37:357–362, 1936). Next, we derive some concentration inequalities for discrete compound Poisson point process (negative binomial random variable with unknown dispersion is a significant example). These concentration inequalities are potentially useful in count data regression. We give an application in the weighted Lasso penalized negative binomial regressions whose KKT conditions of penalized likelihood hold with high probability and then we derive non-asymptotic oracle inequalities for a weighted Lasso estimator.

scholarly journals Asymptotics of Visibility in the Hyperbolic Plane

2013 ◽  
Vol 45 (2) ◽  
pp. 332-350
Author(s):  
Johan Tykesson ◽  
Pierre Calka
Keyword(s):  
Fixed Point ◽  
Point Process ◽  
Hyperbolic Plane ◽  
Poisson Point Process ◽  
Asymptotic Results ◽  
Critical Intensity

At each point of a Poisson point process of intensity λ in the hyperbolic plane, center a ball of bounded random radius. Consider the probability Pr that, from a fixed point, there is some direction in which one can reach distance r without hitting any ball. It is known (see Benjamini, Jonasson, Schramm and Tykesson (2009)) that if λ is strictly smaller than a critical intensity λgv thenPr does not go to 0 as r → ∞. The main result in this note shows that in the case λ=λgv, the probability of reaching a distance larger than r decays essentially polynomially, while if λ>λgv, the decay is exponential. We also extend these results to various related models and we finally obtain asymptotic results in several situations.

Conditional simulation of a Boolean model

1996 ◽  
Vol 28 (02) ◽  
pp. 336
Author(s):  
Ch. Lantuéjoul
Keyword(s):  
Point Process ◽  
Poisson Point Process ◽  
Conditional Simulation ◽  
Boolean Model ◽  
Sequential Algorithm ◽  
Finite Set ◽  
Compact Subsets

A Boolean model in ℝ d is the union of independent random compact subsets of ℝ d (called ‘grains’) located according to a Poisson point process. A sequential algorithm is proposed to conditionally simulate a Boolean model, i.e. so that a finite set of specified points belongs to the grains and another set to the complementary phase (referred to as the ‘pores’).

Critical intensities of Boolean models with different underlying convex shapes

2002 ◽  
Vol 34 (1) ◽  
pp. 48-57 ◽  
Author(s):  
Rahul Roy ◽  
Hideki Tanemura
Keyword(s):  
Unit Area ◽  
Higher Dimensions ◽  
Boolean Model ◽  
Two Dimensions ◽  
Three Dimensions ◽  
Boolean Models ◽  
Regular Polytope ◽  
Minimum Value ◽  
Critical Intensity ◽  
Poisson Boolean Model

We consider the Poisson Boolean model of percolation where the percolating shapes are convex regions. By an enhancement argument we strengthen a result of Jonasson (2000) to show that the critical intensity of percolation in two dimensions is minimized among the class of convex shapes of unit area when the percolating shapes are triangles, and, for any other shape, the critical intensity is strictly larger than this minimum value. We also obtain a partial generalization to higher dimensions. In particular, for three dimensions, the critical intensity of percolation is minimized among the class of regular polytopes of unit volume when the percolating shapes are tetrahedrons. Moreover, for any other regular polytope, the critical intensity is strictly larger than this minimum value.

Conditional simulation of a Boolean model

1996 ◽  
Vol 28 (2) ◽  
pp. 336-336
Author(s):  
Ch. Lantuéjoul
Keyword(s):  
Point Process ◽  
Poisson Point Process ◽  
Conditional Simulation ◽  
Boolean Model ◽  
Sequential Algorithm ◽  
Finite Set ◽  
Compact Subsets

A Boolean model in ℝd is the union of independent random compact subsets of ℝd (called ‘grains’) located according to a Poisson point process. A sequential algorithm is proposed to conditionally simulate a Boolean model, i.e. so that a finite set of specified points belongs to the grains and another set to the complementary phase (referred to as the ‘pores’).

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