Sharpness of the phase transition for continuum percolation in $$\mathbb {R}^2$$ R 2

Abstract

We study the phase transition of random radii Poisson Boolean percolation: Around each point of a planar Poisson point process, we draw a disc of random radius, independently for each point. The behavior of this process is well understood when the radii are uniformly bounded from above. In this article, we investigate this process for unbounded (and possibly heavy tailed) radii distributions. Under mild assumptions on the radius distribution, we show that both the vacant and occupied sets undergo a phase transition at the same critical parameter $$\lambda _c$$ . Moreover, The techniques we develop in this article can be applied to other models such as the Poisson Voronoi and confetti percolation.