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Geometry of the Poisson Boolean model on a region of logarithmic width in the plane. (English) Zbl 1227.60109
Summary: Consider the region $$L = \{(x ,y) : 0 \leq y \leq C\log (1 + x)$$, $$x > 0\}$$ 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 $$\lambda$$ on the entire half space $$\mathbb R_{+}\times\mathbb R$$ and, associated with each Poisson point, we place a box of a random side length $$\rho$$. Depending on the tail behaviour of the random variable $$\rho$$, 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 $$\lambda$$ on the region $$\mathbb R^2$$. At each point of the process, we centre a box of a random side length $$\rho$$. In the case $$\rho \leq R$$ for some fixed $$R>0$$, we study the critical intensity $$\lambda _{c}$$ of the percolation on $$L$$.
##### MSC:
 60K35 Interacting random processes; statistical mechanics type models; percolation theory 60D05 Geometric probability and stochastic geometry
##### Keywords:
Boolean model; Poisson point process; percolation; coverage
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##### References:
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