Existence of Phase Transition for Heavy-Tailed Continuum Percolation

#### A. Sapozhnikov

2007, v.13, Issue 3, 575-586

ABSTRACT

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

Keywords: continuum percolation,percolation,long-range percolation,connectivity,critical density,moments