On the capacity functional of the infinite cluster of a Boolean model

Consider a Boolean model in $\R^d$ with balls of random, bounded radii with distribution $F_0$, centered on the points of a Poisson process of intensity $t>0$. The capacity functional of the infinite cluster $Z_\infty$ is given by $\theta_L(t) = \BP\{ Z_\infty\cap L\ne\emptyset\}$, defined for each compact $L\subset\R^d$. We prove for any fixed $L$ and $F_0$ that $\theta_L(t)$ is infinitely differentiable in $t$, except at the critical value $t_c$; we give a Margulis-Russo type formula for the derivatives. More generally, allowing the distribution $F_0$ to vary and viewing $\theta_L$ as a function of the measure $F:=tF_0$, we show that it is infinitely differentiable in all directions with respect to the measure $F$ in the supercritical region of the cone of positive measures on a bounded interval. We also prove that $\theta_L(\cdot)$ grows at least linearly at the critical value. This implies that the critical exponent known as $\beta$ is at most 1 (if it exists) for this model. Along the way, we extend a result of Tanemura, on regularity of the supercritical Boolean model in $d \geq 3$ with fixed-radius balls, to the case with bounded random radii.