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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
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[1] Athreya, S., Roy, R. and Sarkar, A. (2004). On the coverage of space by random sets. Adv. Appl. Prob. 36, 1-18. · Zbl 1044.60091 · doi:10.1239/aap/1077134461
[2] Grimmett, G. R. (1983). Bond percolation on subsets of the square lattice, and the threshold between one-dimensional and two-dimensional behaviour. J. Phys. A. 16, 599-604. · Zbl 0508.60082 · doi:10.1088/0305-4470/16/3/019
[3] Grimmett, G. R. (1999). Percolation , 2nd edn. Springer, Berlin. · Zbl 0926.60004
[4] Hall, P. (1988). Introduction to the Theory of Coverage Processes . John Wiley, New York. · Zbl 0659.60024
[5] Meester, R. and Roy, R. (1996). Continuum Percolation . Cambridge University Press. · Zbl 0858.60092
[6] Molchanov, I. and Scherbakov, V. (2003). Coverage of the whole space. Adv. Appl. Prob. 35, 898-912. · Zbl 1041.60013 · doi:10.1239/aap/1067436326
[7] Penrose, M. (2003). Random Geometric Graphs . Oxford University Press. · Zbl 1029.60007
[8] Petrov, V. V. (2004). A generalization of the Borel-Cantelli lemma. Statist. Prob. Lett. 67, 233-239. · Zbl 1101.60300 · doi:10.1016/j.spl.2004.01.008
[9] Stoyan, D., Kendall, W. S. and Mecke, J. (1987). Stochastic Geometry and Its Applications . John Wiley, Chichester. · Zbl 0622.60019
[10] Tanemura, H. (1993). Behavior of the supercritical phase of a continuum percolation model on \(\mathbbR^ d\). J. Appl. Prob. 30, 382-396. · Zbl 0778.60070 · doi:10.2307/3214847
[11] Tanemura, H. (1996). Critical behavior for a continuum percolation model. In Probability Theory and Mathematical Statistics. (Tokyo, 1995), World Scientific, River Edge, NJ, pp. 485-495. · Zbl 0961.60095
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