On Continuum Percolation

Abstract

Let $\mathscr{P}$ be a homogeneous Poisson process in $\mathbb{R}^k$. At the points of $\mathscr{P}$, centre $k$-dimensional spheres whose radii are independent and identically distributed. It is shown that there exists a positive critical intensity for the formation of clumps whose mean size is infinite, if and only if sphere content has finite variance. It is also proved that under a strictly weaker condition than existence of finite variance, there exists a positive critical intensity for the formation of clumps whose size is infinite with positive probability. Therefore these two critical intensities need not be the same. Continuum percolation in the case of general random sets, not just spheres, is studied, and bounds are obtained for a critical intensity.