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In a previous work, two of the authors proposed a new proof of a well known convergence result for the scaled elementary connected vacant component in the high intensity Boolean model towards the Crofton cell of the Poisson hyperplane process. In this paper, we consider the particular case of the two-dimensional Boolean model where the grains are discs with random radii. We investigate the second-order term in this convergence when the Boolean model and the Poisson line process are coupled on the same probability space. A precise coupling between the Boolean model and the Poisson line process is first established, a result of directional convergence in distribution for the difference of the two sets involved is derived as well.
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We consider the Bernoulli Boolean discrete percolation model on the d-dimensional integer lattice. We study sufficient conditions on the distribution of the radii of balls placed at the points of a Bernoulli point process for the absence of percolation, provided that the intensity of the underlying point process is small enough. We also study a Harris graphical procedure to construct, forward in time, particle systems with interactions of infinite range under the assumption that the corresponding generator admits a Kalikow-type decomposition. We do so by using the subcriticality of the boolean model of discrete percolation.
The aim of this paper is to give a precise estimate on the tail probability of the visibility function in a germ-grain model: this function is defined as the length of the longest ray starting at the origin that does not intersect an obstacle in a Boolean model. We proceed in two or more dimensions using coverage techniques. Moreover, convergence results involving a type I extreme value distribution are shown in the two particular cases of small obstacles or a large obstacle-free region.
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.
We study the volatility of the output of a Boolean function when the input bits undergo a natural dynamics. For $n = 1,2,ldots$, let $f_n:{0,1}^{m_n} ra {0,1}$ be a Boolean function and $X^{(n)}(t)=(X_1(t),ldots,X_{m_n}(t))_{t in [0,infty)}$ be a vector of i.i.d. stationary continuous time Markov chains on ${0,1}$ that jump from $0$ to $1$ with rate $p_n in [0,1]$ and from $1$ to $0$ with rate $q_n=1-p_n$. Our object of study will be $C_n$ which is the number of state changes of $f_n(X^{(n)}(t))$ as a function of $t$ during $[0,1]$. We say that the family ${f_n}_{nge 1}$ is volatile if $C_n ra iy$ in distribution as $ntoinfty$ and say that ${f_n}_{nge 1}$ is tame if ${C_n}_{nge 1}$ is tight. We study these concepts in and of themselves as well as investigate their relationship with the recent notions of noise sensitivity and noise stability. In addition, we study the question of lameness which means that $Pro(C_n =0)ra 1$ as $ntoinfty$. Finally, we investigate these properties for a number of standard Boolean functions such as the majority function, iterated 3-majority, the AND/OR function on the binary tree and percolation on certain trees at various levels of the parameter $p_n$.
We consider the discrete Boolean model of percolation on graphs satisfying a doubling metric condition. We study sufficient conditions on the distribution of the radii of balls placed at the points of a Bernoulli point process for the absence of percolation, provided that the retention parameter of the underlying point process is small enough. We exhibit three families of interesting graphs where the main result of this work holds. Finally, we give sufficient conditions for ergodicity of the discrete Boolean model of percolation.
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