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.

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Polymat
Markov Processes and Related Fields
Stochastics

Sapozhnikov, A. (2007). Existence of phase transition for heavy-tailed continuum percolation. Markov Processes and Related Fields, 13(3), 575–586.