Theoretical Statistics and Mathematics Unit, ISI Delhi

Geometry of the Poisson Boolean model on a region of logarthmic width in the plane

by Amites Dasgupta, Rahul Roy and Anish Sarkar

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 percolation properties we place a Poisson point process of intensity $\lambda$ on the region $L$. At each point of the process we centre a box of a random side length $\rho$. In case $ \rho \leq R$ for some fixed $R > 0$ we study the critical intensity $\lambda_c$ of percolation.
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 $L$.

isid/ms/2009/04 [fulltext]

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