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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 \(L\).
60K35 Interacting random processes; statistical mechanics type models; percolation theory
60D05 Geometric probability and stochastic geometry
Full Text: DOI
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