| We consider the so-called Poisson Boolean model of continuum percolation. At each point of an homogeneous Poisson point process on the Euclidean space $\R^d$, we center a ball with random radius. We assume that the radii of the balls are independent, identically distributed and independent of the point process. We denote by $\Sigma$ the union of the balls and by $S$ the connected component of $\Sigma$ that contains the origin. We show that $S$ is almost surely bounded for small enough density $\lambda$ of the point process if and only if the mean volume of the balls is finite. Let us denote by $D$ the diameter of $S$ and by $R$ one of the random radii. We also show that, for all positive real number $s$, $D^s$ is integrable for small enough $\lambda$ if and only if $R^{d+s}$ is integrable. |
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