Existence of phase transition for heavy-tailed continuum percolation

We consider a continuum percolation model in $R^d$, where $d >= 2$. It is given by a homogeneous Poisson process of intensity $\labda$ and independent radii random variables of common distribution of a positive random variable $r$. Let $\labda_c$ be the critical intensity for the existence of infinite cluster. We provide conditions for positivity of $\labda_c$. In case $Er^{2d−1} = \infty$ our result is new.