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

Consider the region L := {(x, y) : 0 ≤ y ≤ 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 λ on the region L. At each point of the process we centre a box of a random side length ρ. In case ρ ≤ R for some fixed R > 0 we study the critical intensity λc of percolation. For the coverage properties we place a Poisson point process of intensity λ on the entire half space R+ ×R and associated with each Poisson point we place a box of a random side length ρ. Depending on the tail behaviour of the random variable ρ we exhibit a phase transition in the intensity for the eventual coverage of the region L.